New directions in Ramsey theory∗

New developments in the study of Ramsey theory for graphs are described. In particular, it is discussed how Ramsey theory has evolved from classical Ramsey numbers to more general Ramsey numbers, bipartite Ramsey numbers, k-Ramsey numbers, s-bipartite Ramsey numbers, Ramsey sequences of graphs, and ascending Ramsey indices.


Introduction
During the summer of 1980, Frank Harary was in England where he had arranged to visit Lettice Ramsey, who was perhaps best known for the photography studio Ramsey & Muspratt that she operated with Helen Muspratt. The purpose of Harary's visit with Lettice Ramsey was not photography, however. Mrs. Ramsey had invited Harary to view the numerous papers and files of her late husband Frank Ramsey, who had died some 50 years earlier. Frank Ramsey, who passed away at age 26, had left a large collection of material he had been working on. Ramsey had numerous interests, including philosophy, economics, politics, and mathematics. Indeed, his brother Michael once said about Frank Ramsey: He was interested in almost everything. It was Ramsey's mathematical notes in which Frank Harary was primarily interested, however.
In 1930, the year that Ramsey died, a paper of his titled "On a Problem of Formal Logic" was published in the Proceedings of the London Mathematical Society [28]. This paper contained a result, a restricted version of which is stated below. Theorem 1.1. (Ramsey's Theorem) For any k + 1 ≥ 3 positive integers t, n 1 , n 2 , . . ., n k , there exists a positive integer N such that if each of the t-element subsets of the set {1, 2, . . ., N } is colored with one of the k colors 1, 2, . . ., k, then for some integer i with 1 ≤ i ≤ k, there is a subset S of {1, 2, . . ., N } containing n i elements such that every t-element subset of S is colored i. Each 2-element subset of {1, 2, . . ., N } can then be considered as an edge of the complete graph K N . The most studied case of Ramsey's theorem is the one that occurs by taking k = 2. In this case, only two colors are involved, usually taken to be red and blue. Here, each edge of K N is colored either red or blue, resulting in a red-blue coloring of K N . Now, writing s for n 1 and t for n 2 , Ramsey's theorem becomes the following result.
Inspired by this problem, Robert Greenwood and Andrew Gleason [20] not only showed that R(3, 3) = 6, but showed as well that R(3, 4) = 9, R(3, 5) = 14, and R(4, 4) = 18. In fact, they established an upper bound for the Ramsey numbers R(s, t) for any two positive integers s and t.
The only known Ramsey numbers R(s, t) with 3 ≤ s ≤ t are those stated in the following table. In particular, the Ramsey number R(5, 5) is not known. All that is known about this Ramsey number is 43 ≤ R(5, 5) ≤ 48 (see [4,25]). This means that every red-blue coloring of K 48 results in a red K 5 or a blue K 5 (that is, a monochromatic K 5 ) and there is a red-blue coloring of K 42 for which there is neither a red K 5 nor a blue K 5 .
The Ramsey numbers R(s, t) = R(K s , K t ) have become known as the classical Ramsey numbers. Classical Ramsey numbers are not limited to two positive integers (and two colors), however. For example, for positive integers r, s, and t, the Ramsey number R(r, s, t) is the smallest positive integer n for which every red-blue-green coloring of K n results in either a red K r , a blue K s , or a green K t . As expected, very few such Ramsey numbers R(r, s, t) have been determined for 3 ≤ r ≤ s ≤ t. In fact, only two of these numbers have been found. In 2016, Michael Codish, Michael Frank, Avraham Itzhakov, and Alice Miller [13] showed that R(3, 3, 4) = 30. The Ramsey number R(3, 3, 3) was determined by Greenwood and Gleason in their 1955 paper [20]. The proof we present, however, is due to Sun and Cohen [31]. Proof. First, we show that R(3, 3, 3) ≤ 17. Let there be given a red-blue-green coloring of K 17 . Since the degree of every vertex of K 17 is 16, every vertex is incident with at least six edges of the same color, say the vertex v of K 17 is incident with six green edges vv 1 , vv 2 , . . . , vv 6 . If any two vertices in the set S = {v 1 , v 2 , . . . , v 6 } are joined by a green edge, then K 17 has a green K 3 . On the other hand, if no edge joining two vertices of S is colored green, then every such edge is colored red or blue. Since R(3, 3) = 6, there is a red K 3 or a blue K 3 . Consequently, R(3, 3, 3) ≤ 17.
Next, we show that R(3, 3, 3) ≥ 17. Consider the complete graph G = K 16 whose 16 vertices are labeled with the 16 elements of the additive group Z 4 2 = Z 2 × Z 2 × Z 2 × Z 2 , every element of which is self-inverse. where r, b, g represent the colors red, blue, green, respectively. These three sets have the property that the sum of any two Clearly, there is no monochromatic K 3 in G containing 0000. Suppose that there are three distinct non-identity vertices u, v, w that form a monochromatic K 3 in G.

More general Ramsey numbers
Later, Ramsey numbers more general than the classical Ramsey numbers began to be investigated. For two graphs F and H, not necessarily complete, the Ramsey number R(F, H) is defined as the minimum positive integer n for which every red-blue coloring of K n results in either a subgraph isomorphic to F , all of whose edges are colored red (a red F ) or a subgraph isomorphic to H, all of whose edges are colored blue (a blue H). Frank Harary always liked this notation! All such Ramsey numbers R(F, H) exist as well, for if F has order s and H has order t, then R(F, H) ≤ R(s, t). The dynamic survey "Small Ramsey numbers" by Stanislaw Radziszowski [27] provides a host of information on such Ramsey numbers.
While determining R(F, H) is challenging in most instances, in a paper appearing in the first volume of the Journal of Graph Theory (founded by Frank Harary), Vašek Chvátal [12] found the exact value of R(F, H) whenever F is any tree of a fixed order and H is the complete graph of a fixed order. For the proof of this result, we first present the following known lemma (see [10], for example). Lemma 3.1. Let T be a tree of order k. If G is a graph for which δ(G) ≥ k − 1, then G contains a subgraph isomorphic to T . Proof. We proceed by induction. The result is obvious for k = 1, 2, 3. Assume for an integer k where k ≥ 4 that for every tree T of order k − 1 and every graph G with δ(G ) ≥ k − 2 that G contains a subgraph isomorphic to T . Now, let T be a tree of order k and let G be a graph with δ(G) ≥ k − 1. Let v be an end-vertex of T where u is the neighbor of v. Then T − v is a tree of order k − 1. Let x be a vertex of G and let H = G − x. Since δ(G) ≥ k − 1, we have δ(H) ≥ k − 2. By the induction hypothesis, H contains a subgraph T isomorphic to T − v. Let u be the vertex of T corresponding to u in T − v in an isomorphism. Since deg G u ≥ k − 1, the vertex u is adjacent to a vertex y ∈ V (G) − V (T ). Adding the vertex y and the edge xy to T produces a tree in G isomorphic to T . Theorem 3.1. [12] Let T be a tree of order p ≥ 2. For every integer n ≥ 2, Proof. First, we show that R(T, K n ) ≥ (p − 1)(n − 1) + 1. Let there be given a red-blue coloring of the complete graph K (p−1)(n−1) such that the resulting red subgraph is (n − 1)K p−1 ; that is, the red subgraph consists of n − 1 copies of K p−1 . Since each component of the red subgraph has order p − 1, it contains no connected subgraph of order greater than p − 1.
In particular, there is no red tree of order p. The blue subgraph is then the complete (n − 1)-partite graph K p−1,p−1,...,p−1 , where every partite set contains exactly p − 1 vertices. Hence, there is no blue K n either. Since this red-blue coloring avoids both a red tree T and a blue K n , it follows that R(T, K n ) ≥ (p − 1)(n − 1) + 1.
We now show that R(T, K n ) ≤ (p − 1)(n − 1) + 1 for an arbitrary but fixed tree T of order p ≥ 2 and an integer n ≥ 2. We verify this inequality by induction on n. For n = 2, we show that R(T, K 2 ) ≤ (p − 1)(2 − 1) + 1 = p. Let there be given a red-blue coloring of K p . If any edge of K p is colored blue, then a blue K 2 is produced. Otherwise, every edge of K p is colored red and a red T is produced. Thus, R(T, K 2 ) ≤ p. Therefore, the inequality R(T, K n ) ≤ (p − 1)(n − 1) + 1 holds when n = 2. Assume for an integer k ≥ 2 that R(T, K k ) ≤ (p − 1)(k − 1) + 1. Consequently, every red-blue coloring of K (p−1)(k−1)+1 contains either a red T or a blue K k . We now show that R(T, K k+1 ) ≤ (p − 1)k + 1. Let there be given a red-blue coloring of K (p−1)k+1 . We show that there is either a red tree T or a blue K k+1 . We consider two cases.
. By the induction hypothesis, H contains either a red T or a blue K k . If H contains a red T , so does K (p−1)k+1 . On the other hand, if H contains a blue K k , then, since v is joined to every vertex of H by a blue edge, there is a blue K k+1 in K (p−1)k+1 .
Case 2. Every vertex of K (p−1)k+1 is incident with at most (p−1)(k−1) blue edges. So, every vertex of K (p−1)k+1 is incident with at least p − 1 red edges. Thus, the red subgraph of K (p−1)k+1 has minimum degree at least p − 1. By Lemma 3.1, this red subgraph contains a red T . Therefore, K (p−1)k+1 contains a red T as well.
Since R(3, 3) = R(K 3 , K 3 ) = 6, it follows that R(C 3 , C 3 ) = 6, that is, the complete graphs K 3 could be looked at in terms of 3-cycles. This suggests investigating the Ramsey numbers of other cycles, say R(C 4 , C 4 ) for example. As we mentioned earlier, the complete graph K 5 can be decomposed into two 5-cycles. Once again, coloring one 5-cycle red and the other blue shows that there is a red-blue coloring of K 5 where there is neither a red C 4 nor a blue C 4 and so R(C 4 , C 4 ) ≥ 6. We now show that R(C 4 , C 4 ) ≤ 6 (and so R(C 4 , C 4 ) = 6).
Let there be given a red-blue coloring of K 6 . Since R(3, 3) = 6, there is either a red C 3 or a blue C 3 , say the former. Let C be a red C 3 in K 6 with vertices u, v, w and let x, y, z be the remaining three vertices of K 6 . If any of x, y, z is joined to C by two red edges, then a red C 4 is produced. Thus, we may assume that each of x, y, z is joined to C by at least two blue edges. If any two of x, y, z are joined to the same two vertices of C by blue edges, then a blue C 4 is produced. Thus, we may further assume that the red-blue coloring of K 6 contains the red-blue subgraph shown in Figure 1, where bold edges represent red edges and dashed edges are blue edges. If any of the uncolored edges xy, xz, yz is red, then a red C 4 is produced; if any two of these edges are blue, then a blue C 4 is produced. Hence, in any case, a monochromatic C 4 is produced, which says that R(C 4 , C 4 ) ≤ 6. Consequently, R(C 4 , C 4 ) = 6.
The Ramsey number R(F, H) has been determined when F and H are both paths and when F and H are both cycles. In the case of two paths, the Ramsey number was determined by Gerencsér and Gyárfás [17].
When F and H are both cycles, the Ramsey number was determined by Faudree and Schelp [16] and, independently, by Károlyi and Rosta (see [24,29,30]). When one of F and H is a path and the other a cycle, the Ramsey number was determined by Faudree et al. in [15]. [16,24,29,30] Let p and q be integers with 3 ≤ p ≤ q.
Theorem 3.4. [15] Let m and n be integers with m, n ≥ 2.
if 4 ≤ m ≤ n and m is even if 2 ≤ n ≤ m and m is even.

Bipartite Ramsey numbers
In 1975 Lowell Beineke and Allen Schwenk [5] introduced a new class of Ramsey numbers by considering for a pair F, H of graphs, a red-blue coloring of the regular complete bipartite graph K r,r rather than the complete graph K n , with the goal of obtaining either a red F or a blue H in K r,r . Of course, since K r,r is a bipartite graph, the only graphs F and H for which this is possible are bipartite graphs. Precisely, for two bipartite graphs F and H, the bipartite Ramsey number BR(F, H) of F and H is the smallest positive integer r such that every red-blue coloring of the r-regular complete bipartite graph K r,r results in either a red F or a blue H. We saw in the preceding section that the standard Ramsey number R(C 4 , C 4 ) is 6. Since C 4 = K 2,2 is bipartite, it is reasonable to consider the bipartite Ramsey number BR(C 4 , C 4 ). In particular, we show that BR(C 4 , C 4 ) = 5.
Proof. Since the graph K 4,4 can be decomposed into two copies of C 8 with one copy colored red and the other colored blue (see Figure 2), there is a red-blue coloring of K 4,4 that avoids a monochromatic C 4 . Therefore, BR(C 4 , C 4 ) ≥ 5. It remains to show that BR(C 4 , C 4 ) ≤ 5. Let there be given a red-blue coloring of K 5,5 whose partite sets are denoted by U and W . The partite set U , for example, contains three vertices, each incident with three or more edges of the same color, say red edges. Then two of these vertices are joined to two vertices of W by red edges, producing a red C 4 . Thus, BR(C 4 , C 4 ) ≤ 5 and so BR(C 4 , C 4 ) = 5.
While BR(C 4 , C 4 ) = 5, the primary question here is that of determining those bipartite graphs F and H for which BR(F, H) exists. It turns out in fact that BR(F, H) exists for every pair F, H of bipartite graphs. Although this was stated in [5], an upper bound for BR(F, H) was obtained by Johann Hattingh and Michael Henning [21], thereby establishing this existence result. In order to state this bound, it is useful to introduce some additional terminology.
For positive integers s and t, bipartite Ramsey numbers of the type BR(K s,s , K t,t ) are referred to as classical bipartite Ramsey numbers. These numbers are also denoted by BR(s, t). Showing that BR(s, t) exists for every pair s, t of positive integers shows that BR(F, H) exists for every pair F, H of bipartite graphs. To see this, let F and H be two bipartite graphs, where the largest partite set of F has s vertices and the largest partite set of H has t vertices. Then BR(F, H) ≤ BR(s, t).
The following result of Hattingh and Henning [21] is analogous to Theorem 2.1.
Theorem 4.1. [21] For every two positive integers s and t, According to the bound for BR(s, t) given in Theorem 4.1, We have already seen in Example 4.1 that BR(C 4 , C 4 ) = 5, so this bound is attained when s = t = 2. It was shown in [5] that BR(2, 3) = 9 and BR(2, 4) = 14, so this bound is also attained when s = 2 and t ∈ {3, 4}. However, it was also shown in [5] that BR(3, 3) = 17 and so the bound is not attained when s = t = 3. This last bipartite Ramsey number then gives the answer to the question asked in the following "party problem": Suppose, for some positive integer r, that r girls and r boys are invited to a party where each girl-boy pair are either acquainted or are strangers. What is the smallest such r that guarantees that there exists a group of six people, three girls and three boys, such that either (1) every one of the three girls is acquainted with every one of the three boys or (2) every one of the three girls is a stranger of every one of the three boys?
Bipartite Ramsey numbers can be defined for more than two bipartite graphs. For example, BR(s, t, p) is the smallest positive integer r for which any red-blue-green coloring of K r,r results in either a red K s,s , a blue K t,t , or a green K p,p . That these numbers exist (as well as for any prescribed number k of bipartite graphs and colors) is a consequence of a theorem of Paul Erdős and Richard Rado [14]. The only nontrivial bipartite Ramsey number that has been determined for k ≥ 3 is BR(2, 2, 2) = BR(K 2,2 , K 2,2 , K 2,2 ) = 11, a result due to Goddard, Henning, and Oellermann [18].

k-Ramsey numbers
If, for two bipartite graphs F and H, we have BR(F, H) = r, then it follows that for every red-blue coloring of K r,r , there is either a red F or a blue H; while there exists a red-blue coloring of K r−1,r−1 for which there is neither a red F nor a blue H. This brings up the question of which situation can occur for the graph K r−1,r . This question led to a concept introduced in [3].
For bipartite graphs F and H, the 2-Ramsey number R 2 (F, H) is the smallest positive integer n such that every red-blue coloring of the complete bipartite graph K n/2 , n/2 of order n results in a red F or a blue H. In particular, if BR(F, H) = r and there is a red-blue coloring of K r−1,r that avoids both a red F and a blue H, then R 2 (F, H) = 2r; while if every red-blue coloring of K r−1,r produces either a red F or a blue H, then R 2 (F, H) = 2r − 1. In general then, either We saw in Example 4.1 that BR(C 4 , C 4 ) = 5. This implies that either R 2 (C 4 , C 4 ) = 10 or R 2 (C 4 , C 4 ) = 9.
Proof. Let K 2,3 be the complete bipartite graph where u 1 , u 2 , u 3 are the three vertices of degree 2 and let H be the graph obtained from K 2,3 by subdividing each edge incident with u 1 or u 2 exactly once. The graph H is shown Figure 3(c). Then H does not contain C 4 as a subgraph. Since K 4,5 can be decomposed into two copies of H, with one copy colored red (shown in Figure 3(a)) and the other copy colored blue (shown in Figure 3(b)), it follows that R 2 (C 4 , C 4 ) = 9. Therefore, R 2 (C 4 , C 4 ) = 10 by (1). There is a concept even more general than the 2-Ramsey number of bipartite graphs. For an integer k ≥ 2, a balanced complete k-partite graph of order n ≥ k is the complete k-partite graph in which every partite set has either n/k or n/k vertices. In particular, if n = kq + r (by the Division Algorithm) where q ≥ 1 and 0 ≤ r ≤ k − 1, then the balanced complete k-partite graph G of order n has r partite sets with q + 1 vertices and k − r partite sets with q vertices. If r = 0, then G is a (k − 1)q-regular graph.
For bipartite graphs F and H and an integer k with 2 ≤ k ≤ R(F, H), the k-Ramsey number R k (F, H) is defined as the smallest positive integer n such that every red-blue coloring of a balanced complete k-partite graph of order n results in a red F or a blue H. That is, R k (F, H) is the minimum order of a balanced complete k-partite graph every red-blue coloring of which results in a red F or a blue H.
If F and H are two bipartite graphs for which R(F, H) = n ≥ 3, then every red-blue coloring of K n produces either a red F or a blue H. However, such is not the case for the smaller complete graphs K 2 , K 3 , . . ., K n−1 . Equivalently, for every red-blue coloring of the complete n-partite graph K n where each partite set consists of a single vertex, there is either a red F or a blue H. However, for each complete k-partite graph K k , where 2 ≤ k ≤ n − 1 such that every partite set consists of a single vertex, there exists a red-blue coloring that produces neither a red F nor a blue H. On the other hand, for each of the graphs K 2 , K 3 , . . ., K n−1 , we can continue to add vertices to each partite set, resulting in a balanced complete k-partite graph at each step where 2 ≤ k ≤ n − 1 until eventually arriving at the balanced complete k-partite graph of smallest order R k (F, H) having the property that every red-blue coloring of this graph produces a red F or a blue H. Consequently, for every two bipartite graphs F and H and every integer k with 2 ≤ k ≤ R (F, H), the k-Ramsey number R k (F, H) exists. Furthermore, if R(F, H) = n, then R n (F, H) ≤ R n−1 (F, H) ≤ · · · ≤ R 3 (F, H) ≤ R 2 (F, H).
While the k-Ramsey number R k (F, H) exists for every two bipartite graphs F and H when 2 ≤ k ≤ R (F, H), such is not the case when F and H are not bipartite. For graphs F and H that are not bipartite, not only does R 2 (F, H) fail to exist but R 3 (F, H) and R 4 (F, H) also do not exist. To see this, let G be any balanced complete 3-partite graph with partite sets V 1 , V 2 , and V 3 . Assigning the color red to every edge of [V 1 , V 2 ], the set of edges joining a vertex of V 1 and a vertex of V 2 , and blue to all other edges of G results in a red subgraph G R and a blue subgraph G B both being bipartite. Similarly, if G is a balanced complete 4-partite graph with partite sets V 1 , V 2 , V 3 , and V 4 , where the color red is assigned to every edge V 4 ] and the color blue to all other edges of G, then G R and G B are both bipartite. Indeed, even if χ(F ) = χ(H) = 3, R 5 (F, H) need not exist. For example, R 5 (K 3 , K 3 ) does not exist. To see this, let G be a balanced complete 5-partite graph with partite sets are colored red and all other edges are colored blue, then G does not contain a monochromatic K 3 . Consequently, R k (K 3 , K 3 ) exists only when k = R(K 3 , K 3 ) = 6. Even if a red-blue coloring of K 5 does not contain a monochromatic K 3 , there is another monochromatic graph it must contain.
Observation 5.2. Every red-blue coloring of K 5 produces either a monochromatic C 3 or a monochromatic C 5 .
From our preceding discussion, the k-Ramsey number of two odd cycles does not exist when k = 2, 3, 4. Furthermore, the 5-Ramsey number of two triangles does not exist. However, if neither of the two odd cycles is a triangle, then the situation is different. The following was shown in [2].
We have seen that Ramsey numbers are defined for three or more graphs. In particular, for three graphs F 1 , F 2 , and F 3 , the Ramsey number R(F 1 , F 2 , F 3 ) is the smallest positive integer n for which every red-blue-green coloring (in which every edge is colored red, blue, or green) of the complete graph K n results in a red F 1 , a blue F 2 , or a green F 3 . This gives rise to the concept of the k-Ramsey number of three (or more) graphs. For three graphs F 1 , F 2 , and F 3 and an integer k with 2 ≤ k ≤ R(F 1 , F 2 , F 3 ), the k-Ramsey number R k (F 1 , F 2 , F 3 ), if it exists, is the smallest order of a balanced complete k-partite graph G for which every red-blue-green coloring of G results in a red F 1 , a blue F 2 , or a green F 3 . In particular, if k = 2 and F i ∼ = F for some graph F , where i = 1, 2, 3, then the 2-Ramsey number R 2 (F, F, F ) is the smallest order of a balanced complete bipartite graph G for which every red-blue-green coloring of G results in a monochromatic F . For example, we mentioned that it was shown in [18] that BR(C 4 , C 4 , C 4 ) = 11, implying that R 2 (C 4 , C 4 , C 4 ) ≥ 21. Furthermore, it was shown in [22] that R 2 (C 4 , C 4 , C 4 ) ≤ 21. Consequently, R 2 (C 4 , C 4 , C 4 ) = 21.

s-Bipartite Ramsey numbers
In the two preceding sections, we have seen that BR(C 4 , C 4 ) = 5 and R 2 (C 4 , C 4 ) = 10. From this, it follows that every red-blue coloring of K 5,5 results in a monochromatic C 4 , while there exists a red-blue coloring of K 4,5 that avoids a monochromatic C 4 . This brings up another question. Does every red-blue coloring of K 4,6 produce a monochromatic C 4 or is there some red-blue coloring of K 4,6 that avoids a monochromatic C 4 ? This can be answered by observing that (1) the graph H in Figure 4 does not contain a 4-cycle and (2) the graph K 4,6 can be decomposed into two copies of H (or K 4,6 is H-decomposable). Thus, by coloring one copy of H red and the other blue, a red-blue coloring of K 4,6 is produced that avoids a monochromatic C 4 . This changes with the graph K 4,7 , however. In fact, not only does every red-blue coloring of K 4,7 contain a monochromatic C 4 , every red-blue coloring of K 3,7 contains a monochromatic C 4 . To see this, let there be given a red-blue coloring of G = K 3,7 resulting in a red subgraph G R and a blue subgraph G B , the sizes of which are denoted by m R and m B , respectively. Since the size of G is 21, one of m R and m B is at least 11, say m R ≥ 11. Let U and W be the partite sets of G, where |U | = 3 and |W | = 7. If U contains vertices u 1 and u 2 such that deg G R u 1 + deg G R u 2 ≥ 9, then u 1 and u 2 have two common neighbors in G R and so G R contains a 4-cycle. Otherwise, the degrees of the three vertices of U in G R are either 5, 3, 3, or 4, 4, 4, or 4, 4, 3. In any of these three cases, two vertices of U have two common neighbors in G R , resulting in a 4-cycle in G R .
These observations resulted in a concept introduced in [7]. For two bipartite graphs F and H and a positive integer s, the s-bipartite Ramsey number BR s (F, H) of F and H is the smallest integer t with t ≥ s such that every red-blue coloring of K s,t results in a red F or a blue H. From our discussion above, we have the following result. Here, we write BR s (K p,p , K q,q ) as BR s (p, q). Theorem 6.1. [7] For each integer s ≥ 2, Proof. First, let t ≥ 2 be an integer and let G = K 2,t , where {u 1 , u 2 } is one of the partite sets of G. If each edge of G incident with u 1 is colored red and each edge incident with u 2 is colored blue, then there is no monochromatic K 2,2 . Thus, BR 2 (2, 2) does not exist. We have seen that every red-blue coloring of K 3,7 has a monochromatic C 4 = K 2,2 and there exists a red-blue coloring of K 4,6 that avoids a monochromatic K 2,2 . Therefore, BR 3 (2, 2) = BR 4 (2, 2) = 7.
There are five girls at a party. What is the minimum number of boys who must be invited to the party to guarantee that there exists a group of six people, three girls and three boys, such that either (1) every one of the three girls is acquainted with every one of the three boys or (2) every one of the three girls is a stranger of every one of the three boys?

Ramsey sequences
The establishment of the existence of the classical Ramsey numbers R(s, s), indirectly by Ramsey, and the classical bipartite Ramsey numbers of BR(s, s) by Beineke and Schwenk for every positive integer s suggested another Ramsey concept (and another class of problems) stated in [10, p. 313] and [11]. A sequence {G k } of graphs is ascending if G k is isomorphic to a proper subgraph of G k+1 for every positive integer k. Furthermore, an ascending sequence {G k } of graphs is a Ramsey sequence if for every positive integer k, there is an integer n > k such that every red-blue coloring of G n results in either a red G k or a blue G k , that is, a monochromatic G k . The theorems obtained by Ramsey and by Beineke and Schwenk show, respectively, that {K k } and {K k,k } are both Ramsey sequences.
Even though {K k,k,k } is an ascending sequence, it is not a Ramsey sequence. To see this, let k be a given integer and let n be an integer where n > k. Let the partite sets of K n,n,n be V 1 , V 2 , V 3 . Color each edge of [V 1 , V 2 ] red and color all remaining edges of K n,n,n blue. Then every monochromatic subgraph of K n,n,n is bipartite, while K k,k,k is not. Similarly, if {G k } is any ascending sequence for which {χ(G k )} is a constant sequence of an integer 3 or more, then {G k } is not a Ramsey sequence. This results in the following. Proposition 7.1. [11] If {G k } is a Ramsey sequence, then either every graph G k is bipartite or lim k→∞ χ(G k ) = ∞.
Proof. Since the sequence {G k } of graphs is ascending, it follows that χ(G k+1 ) ≥ χ(G k ) for every positive integer k. Assume that neither every graph G k is bipartite nor lim k→∞ χ(G k ) = ∞. Therefore, there is a positive integer N such that for every integer k ≥ N , it follows that χ(G k ) is a constant p ≥ 3. Let n be an arbitrary integer with n > N . Then χ(G n ) = p. Let V 1 , V 2 , . . . , V p be the color classes in a proper p-coloring of the vertices of G n . Assign the color red to all edges in V p , p−1 i=1 V i and the color blue to the remaining edges of G n . Since the resulting red subgraph of G n is bipartite and the chromatic number of the blue subgraph of G n is p − 1, it follows that there is no monochromatic subgraph of G n that is isomorphic to G N . Hence, {G k } is not a Ramsey sequence.
While {K r,r } is a Ramsey sequence of bipartite graphs, {K k } is a Ramsey sequence for which lim For a graph G k in a Ramsey sequence S = {G k } of graphs, the smallest positive integer n for which every red-blue coloring of G n results in a monochromatic G k is referred to as the S-Ramsey number R S (G k ) of If S = {K r,r }, then R S (K s,s ) ≤ 2s s − 1.
Another Ramsey sequence of graphs is that of the stars. Proof. For a positive integer t, every red-blue coloring of K 2t−1 produces either a red K 1,t or a blue K 1,t and so R S (K 1,t ) ≤ 2t − 1. Since the red-blue coloring of H = K 2t−2 that assigns red to t − 1 edges of H and blue the remaining t − 1 edges of H avoids a red K 1,t and a blue K 1,t , it follows that R S (K 1,t ) ≥ 2t − 1 and so R S (K 1,t ) = 2t − 1.
Another simple Ramsey sequence consists of disconnected graphs. An argument similar to the one in the proof of Proposition 7.2 gives the following result.  Proposition 7.4. [11] If H is any connected graph of order 3 or more, then {kH} is not a Ramsey sequence.
Proof. The sequence S = {kH} is clearly an ascending sequence. For an integer k, let G = kH and let H 1 , H 2 , . . . , H k be the k vertex-disjoint copies of H in G. Define a red-blue coloring of G by assigning red to one edge of H i and blue to the remaining edges of H i for 1 ≤ i ≤ k. Since this coloring avoids a monochromatic H in G, it follows that {kH} is not a Ramsey sequence.
The n-cube or hypercube Q n is K 2 if n = 1, while for n ≥ 2, Q n is defined recursively as the Cartesian product Q n−1 K 2 of Q n−1 and K 2 . The n-cube can also be defined as that graph whose vertex set is the set of n-bit strings (a 1 , a 2 , . . . , a n ) or a 1 a 2 · · · a n , where a i is 0 or 1 for 1 ≤ i ≤ n such that two vertices are adjacent if and only if the corresponding n-bit strings differ at precisely one coordinate. Even though {Q k } is an ascending sequence of bipartite graphs, {Q k } is not a Ramsey sequence.
Theorem 7.2. [11] The sequence {Q k } of hypercubes is not a Ramsey sequence.
Proof. Let S = {Q k }. Since Q k is a proper subgraph of Q k+1 for every positive integer k, it follows that S is ascending. We show that for every integer k ≥ 3, there is a red-blue coloring of Q k that avoids a monochromatic Q 2 = C 4 .
Let -bit string (a 1 , a 2 , . . . , a k ) where exactly i of the coordinates a 1 , a 2 , . . . , a n are 1 and the remaining n − i coordinates are 0. Thus, V 0 = {(0, 0, . . . , 0)} and V k = {(1, 1, . . . , 1) We show that H i contains no C 4 for 0 ≤ i ≤ k − 1. This is clear for H 0 and H k−1 . Assume, to the contrary, that there is an integer i with 1 ≤ i ≤ k − 2 such that H i contains a 4-cycle (w, x, y, z, w), where w, y ∈ V i and x, z ∈ V i+1 . Since wx, wz ∈ E(H i ), there are two coordinates of the vertex w that are both 0, say coordinates p and q, where in x the coordinate p is 1 and in z the coordinate q is 1; while otherwise, the coordinates of w and x and of w and z are identical. This implies, however, that there are two coordinates p and q in x and z, where coordinate p is 1 and the coordinate q is 0 in x, while the coordinate p is 0 and the coordinate q is 1 in z such that changing each 1 to 0 results in the vertex y. However, the only coordinates where this can occur is when p = p and q = q , which implies that y = w. This is impossible.
We now define a red-blue coloring Q k by assigning red to each edge in H i if i is even and 0 ≤ i ≤ k − 1 and blue to each edge in H i if i is odd and 1 ≤ i ≤ k − 1. This red-blue coloring is shown in Figure 5 for Q 4 , where each dashed line is a red edge and a solid line is a blue edge. Since there is no C 4 in H i for 0 ≤ i ≤ k − 1, this red-blue coloring of Q k avoids a monochromatic C 4 . In fact, the only 4-cycles are of the form (w, x, y, z, w), where w ∈ V i , x, z ∈ V i+1 , and y ∈ V i+2 in which case, w has two coordinates p and q, both 0, y has coordinates p and q, both 1, and is otherwise identical to w; while x and z have exactly one of coordinates p and q to be 1 and is otherwise identical to w. This 4-cycle is not monochromatic, however. Therefore, Q k has no monochromatic C 4 ; thus, S is not a Ramsey sequence. We saw in Proposition 7.1 that if {G k } is a Ramsey sequence, then either every graph G k is bipartite or lim We have seen that if {G k } is an ascending sequence of bipartite graphs, then {G k } may or may not be a Ramsey sequence. We now consider ascending sequences {G k } for which lim Proof. Let G j be an arbitrary graph in the sequence {G k } and let R(G j , G j ) = n. Since lim k→∞ ω(G k ) = ∞, there exists an integer p such that for every integer k ≥ p, it follows that ω(G k ) ≥ n and so H = K n is a subgraph of G k . For every red-blue coloring of G k , there exists a red-blue coloring of the subgraph H in G k . Since R(G j , G j ) = n, it follows that there exists a monochromatic G j in H and so there is a monochromatic G j in G k . Therefore, {G k } is a Ramsey sequence.
There are sequences {G k } of graphs for which lim k→∞ χ(G k ) = ∞ and lim k→∞ ω(G k ) = ∞. The question is whether there are sequences with these properties that are ascending and, if so, whether these sequences are Ramsey sequences. In [11] one such sequence was described.
Let G be a graph of order n with V (G) = {u 1 , u 2 , . . . , u n }. Applying a construction (called the Mycielski construction) to G, we obtain a graph, denoted by M (G), of order 2n + 1 by adding a vertex-disjoint star K 1,n to G, where the central vertex of K 1,n is v and the end-vertices are v 1 , v 2 , . . . , v n . Edges are then added between V (G) and {v 1 , v 2 , . . . , v n } so that N (v i ) = N G (u i ) ∪ {v} for i = 1, 2, . . . , n. Here, the vertex v i is often referred to as the shadow vertex of u i . This construction was introduced by Jan Mycielski [26]. If G is triangle-free, then M (G) is also triangle-free and χ(M (G)) = χ(G) + 1. Thus, by successively applying the Mycielski construction, we obtain a sequence G, M (G), M (M (G)) = M 2 (G), . . . of triangle-free graphs where lim k→∞ χ(M k (G)) = ∞. For G = K 2 , the graphs M (G) = C 5 and M 2 (G) are shown in Figure 6. The graph M 2 (G) is the famous Grötzsch graph, which is the triangle-free graph of smallest order that has chromatic number 4. Figure 6: The graphs C 5 and Grötzsch graph.
The Mycielski construction can be applied as well to graphs that are not triangle-free. In particular, we can apply the Mycielski construction to the triangle K 3 . Therefore, . is a sequence of K 4 -free graphs and so ω(G k ) = 3 for every nonnegative integer k with lim k→∞ χ(G k ) = ∞. Figure 7 shows red-blue colorings of K 3 and M (K 3 ), where a red edge is denoted by a dashed line and a blue edge by a solid line. In both red-blue colorings, there is no monochromatic K 3 . The following result was obtained in [11].

Monochromatic ascending subgraph sequences
There is a problem in Ramsey theory that involves both ascending sequences of graphs and graph decompositions. A graph G of size n+1 2 for some integer n ≥ 2 is said to have an ascending subgraph decomposition (ASD) if there exists an ascending sequence {G k } = {G 1 , G 2 , . . ., G n } of n subgraphs of G such that {G 1 , G 2 , . . ., G n } is a decomposition of G where G i has size i for 1 ≤ i ≤ n. This concept and the following conjecture were introduced in [1].
The Ascending Subgraph Decomposition Conjecture. For n ≥ 2, every graph of size n+1 2 has an ascending subgraph decomposition.
Let G be a graph of size m where n+1 2 ≤ m < n+2 2 . The ascending Ramsey index AR(G) of G is the maximum integer k such that for every red-blue coloring of G, there exists an ascending subgraph sequence G 1 , G 2 , . . ., G k such that G i is monochromatic for 1 ≤ i ≤ k. A graph G of size n+1 2 is said to have a monochromatic ascending subgraph decomposition (or a monochromatic ASD) if for every red-blue coloring of G, there exists an ascending subgraph decomposition G 1 , G 2 , . . ., G n of G such that each subgraph G i is monochromatic for 1 ≤ i ≤ n. Consequently, if a graph G of size n+1 2 has a monochromatic ASD, then AR(G) = n. This concept is illustrated in the next two examples.
Example 8.1. The graph K 4 has a monochromatic ASD and so AR(K 4 ) = 3.
Proof. Let there be given an arbitrary red-blue coloring of K 4 , resulting in the red subgraph G R and the blue subgraph G B of sizes m R and m B , respectively, where m R ≤ m B . We show that K 4 has a monochromatic ASD. Since m R ≤ m B , it follows that 0 ≤ m R ≤ 3. For m R ∈ {0, 1}, such an ASD is clear. Suppose that m R = 2. Then either G R = 2K 2 or G R = P 3 . In either case, there is a monochromatic ASD with G 1 = K 2 , G 2 = G R , and G 3 = P 4 . If m R = 3, then G R ∈ {K 3 , K 1,3 , P 4 }. In each case, G 1 = K 2 , G 2 = K 1,2 , G 3 = G R is a monochromatic ASD of K 4 . Example 8.2. The graph G = 3K 2 + K 1,7 of size 10 has ascending Ramsey index 3.
Proof. First, consider the red-blue coloring of G, resulting in the red subgraph G R ∼ = 4K 2 . We show that there is no monochromatic ASD of G into four graphs G 1 , G 2 , G 3 , G 4 of G with this red-blue coloring, for suppose that there is. Then either (1) only G 4 is a red subgraph or (2) only G 1 and G 3 are red subgraphs. We consider these two cases.
Case 1. Only G 4 is a red subgraph of G. Since G 4 = 4K 2 , it follows that G 3 = K 1,3 . Because K 1,3 is not isomorphic to a subgraph of G 4 , this is a contradiction.
Case 2. Only G 1 and G 3 are red subgraphs of G. Since G 1 = K 2 and G 3 = 3K 2 , it follows that G 2 = K 1,2 and G 4 = K 1,4 . Because G 3 is not isomorphic to a subgraph of G 4 , for example, this is a contradiction.
Therefore, AR(G) ≤ 3. It remains to show that AR(G) ≥ 3. Let there be given an arbitrary red-blue coloring of G. Let G 1 = K 2 be any of the three components of size 1 in G and let G 2 = K 1,2 be a monochromatic subgraph of K 1,7 . The remaining subgraph K 1,5 of K 1,7 has three edges colored the same. Let G 3 = K 1,3 be such a monochromatic subgraph of K 1,5 . Then G 1 , G 2 , G 3 is a monochromatic ascending subgraph sequence in G. Thus, AR(G) ≥ 3 and so AR(G) = 3.
If G is a star or a matching of size n+1 2 , then G has a monochromatic ASD and consequently AR(G) = n, as we show next. Proof. We proceed by induction on n. The truth of this statement is immediate for n = 2. Assume for an arbitrary integer n ≥ 2 that every star of size n+1 2 has a monochromatic ASD. Let G be a star of size n+2 2 and let there be given a red-blue coloring of G. Since n ≥ 2, it follows that 1 2 n+2 2 ≥ n + 1 and so there is a monochromatic substar H of G having size n + 1. Let U be the set of the n + 1 end-vertices of the substar H and let G = G − U . Thus, G is a star of size n+1 2 . By the induction hypothesis, the resulting red-blue coloring of G has a monochromatic ASD into n monochromatic subgraphs G 1 , G 2 , . . ., G n . Hence, G 1 , G 2 , . . ., G n , G n+1 = H is a monochromatic ASD of G.
The following result has a proof similar to that of Proposition 8.1. Among the numerous problems on this topic are the following. In addition to stars and matchings of size n+1 2 , which graphs G of size n+1 2 have AR(G) = n? For which positive integers k, do there exist a positive integer n and a graph G of size n+1 2 such that AR(G) = n − k?