Multicolor Ramsey numbers and restricted Tur\'an numbers for the loose 3-uniform path of length three

Let $P$ denote a 3-uniform hypergraph consisting of 7 vertices $a,b,c,d,e,f,g$ and 3 edges $\{a,b,c\}, \{c,d,e\},$ and $\{e,f,g\}$. It is known that the $r$-colored Ramsey number for $P$ is $R(P;r)=r+6$ for $r=2,3$, and that $R(P;r)\le 3r$ for all $r\ge3$. The latter result follows by a standard application of the Tur\'an number $ex_3(n;P)$, which was determined to be $\binom{n-1}2$ in our previous work. We have also shown that the full star is the only extremal 3-graph for $P$. In this paper, we perform a subtle analysis of the Tur\'an numbers for $P$ under some additional restrictions. Most importantly, we determine the largest number of edges in an $n$-vertex $P$-free 3-graph which is not a star. These Tur\'an type results, in turn, allow us to confirm the formula $R(P;r)=r+6$ for $r\in\{4,5,6,7\}$.


Introduction
In this paper we prove results about both Ramsey numbers and Turán numbers for the loose 3-uniform path of length 3 defined as the hypergraph P := P 3 3 consisting of 7 vertices, say, a, b, c, d, e, f, g, and 3 edges {a, b, c}, {c, d, e}, and {e, f, g}. This is a very special case of a more general notion of the k-uniform loose path P k m of length m, where k, m 2, defined as a k-uniform hypergraph (or k-graph, for short) with m edges which can be linearly ordered in such a way that every two consecutive edges intersect in exactly one vertex while all other pairs of edges are disjoint. Note that some authors, e.g., in [5,13] call such paths linear, while by loose they mean paths in which consecutive edges may intersect on more vertices.
The complete k-graph K k n is a k-graph on n vertices in which every k-element subset of the vertex set forms an edge. For a given k-graph F and an integer r 2, the Ramsey number R(F ; r) is the least integer n such that every r-coloring of the edges of K k n results in a monochromatic copy of F . In the classical case of two colors (r = 2), it is known already that for graphs (k = 2) R(P 2 m ; 2) [6] = 3m + 1 2 , while for 3-graphs R(P 3 m ; 2) [14] = 5m + 1 2 , both formulae holding for all m 2. For higher dimensions (k 4), only the numbers R(P k m ; 2), m = 2, 3, 4, have been determined exactly (see [7]), while in [8] an asymptotic formula R(P k m ; 2) ∼ (k − 1/2)m, k fixed, m → ∞, was established. For more than two colors, the only existing results are R(P ; 3) = 9 and r + 6 R(P ; r) 3r for r 3 [10,11]. We include below a simple proof of the upper bound to recall the standard technique of using Turán numbers for bounding Ramsey numbers, as this is the starting point of the research presented in this paper.
For a given k-graph F and an integer n 1, the Turán number ex k (n; F ) is the largest number of edges in an n-vertex F -free k-graph (for a more general definition, see Section 2.) Every n-vertex F -free k-graph with ex k (n; F ) edges is called extremal.
Clearly, if n k > r · ex k (n; F ), then R(F ; r) n. This trivial observation can sometimes be sharpened, owing to a specific structure of the extremal k-graphs. A star is a hypergraph with a vertex, called the center, contained in all the edges. An n-vertex k-uniform star is called full and denoted by S k n if it has n−1 k−1 edges. It has been proved in [11] that for n 8, ex 3 (n; P ) = n−1 2 and that S 3 n is the only extremal 3-graph. Thus, the above inequality is equivalent to n > 3r and yields only that R(P ; r) 3r + 1. If n = 3r, then n 3 = r · ex 3 (n; P ), meaning that for every r-coloring of K 3 n either there is a monochromatic copy of P or every color forms a full star which, however, is impossible. This was good enough to claim that R(P ; 3) = 9 in [10], but for r = 4 it only yielded the bound R(P ; 4) 12. To make further progress in pin-pointing the Ramsey numbers R(P ; r) one has to refine the analysis of the Turán numbers and extremal 3-graphs for P which, in our opinion, might be of independent interest.
Let us illustrate our approach by sticking to the case r = 4 for a while. The lower bound on R(P ; 4) is r + 6 = 10 and 1 4 10 3 = 30 < 9 2 . This only tells us that in every 4-coloring of K 3 10 a color must have been applied to at least 30 edges. If we only knew that the edges of that color formed a star (not necessarily full), then we could remove the center of that star reducing the picture to a 3-coloring of K 3 9 about which we already know that it does contain a monochromatic copy of P .
In this paper we prove that this is, indeed, the case. In fact, we prove a much stronger result by determining precisely the largest number of edges in an n-vertex P -free 3-graph which is not a subset of a star. We call this the Turán number of the second order. This approach works fine for r = 5 and r = 7, but, quite surprisingly, fails for r = 6. In this case, we need to define the Turán number of the third order and compute it for n = 12.
Our contribution to the Ramsey theory of hypergraphs is summarized in the following result.
In the next section we define Turán numbers of the s-th order, s 1, as well as, conditional Turán numbers, and state several results about them with respect to the path P . Then in Section 3, using some of these results, we prove Theorem 1. The remaining sections are all devoted to proving the Turán-type theorems from Section 2.

Turán numbers
In this section, after providing some background, we define Turán numbers of the s-th order as well as conditional Turán numbers, and formulate our results concerning such numbers for P , the loose 3-uniform path of length 3. We begin by recalling the definition of the ordinary Turán number. Given a family of k-graphs F, we call a k-graph H F-free if for all F ∈ F we have F H. Definition 1. For a family of k-graphs F and an integer n 1, the Turán number (of the 1st order) is defined as ex (1) k (n; F) := ex k (n; F) = max{|E(H)| : |V (H)| = n and H is F-free}.
Every n-vertex F-free k-graph with ex k (n; F) edges is called extremal (1-extremal) for F. We denote by Ex k (n; F) = Ex (1) k (n; F) the family of all n-vertex k-graphs which are extremal for F.
In the case when F = {F }, we will often write ex k (n; F ) for ex k (n; {F }) and Ex k (n; F ) for Ex k (n; {F }).
The Turán numbers for graphs have been harder to grasp in the case of bipartite F than when χ(F ) 3. For k-graphs, k 3, on the other hand, the k-partite case seems to be easier. Indeed, the numbers ex k (n; F ) have been already computed for F being a pair of disjoint edges, a loose path and a loose cycle, while, e.g., ex 3 (n; K 3 4 ) is still not known, even asymptotically. Interestingly, the three k-partite cases of F mentioned above exhibit a whole lot of similarity.
A family F of sets is called intersecting if e ∩ e = ∅ for all e, e ∈ F . Obviously, a star is intersecting. Restricting to n-vertex k-graphs, a celebrated result of Erdős, Ko, and Rado asserts that for n 2k + 1, the full star S k n is, indeed, the unique largest intersecting family. Below, we formulate this result in terms of the Turán numbers. Let M k 2 be a k-graph consisting of two disjoint edges.

Theorem 2 ([2]
). For n 2k, ex k (n; M k 2 ) = n−1 k−1 . Moreover, for n 2k + 1, Ex k (n; M k 2 ) = {S k n }. A loose cycle C k m is defined in the same way as a loose path P k m , except that this time also the first and the last edge share one vertex. When k = m = 3 it is sometimes called a triangle. For convenience we abbreviate our notation for triangles to C := C 3 3 . The Turán number ex 3 (n; C) has been determined in [4] for n 75 and later for all n in [1].
). For n 6, ex 3 (n; C) = n−1 2 . Moreover, for n 8, Ex 3 (n; C) = {S 3 n }. Finally, we return to loose paths. For large n, the Turán number for P k m has been determined for k 4 in [5] and for m 4 in [13]. In [5] the authors admitted that their method does not quite work for k = 3, while the authors of [13] credited [5] with that case. In [11] we closed this gap. Given two k-graphs F 1 and F 2 , by F 1 ∪ F 2 we denote a vertex-disjoint union of F 1 and F 2 . Also, note that K 3 1 is just an isolated vertex. and Ex 3 (n; P ) = {K 3 n } for n 6, 20 and Ex 3 (n; P ) = {K 3 6 ∪ K 3 1 } for n = 7, n−1 2 and Ex 3 (n; P ) = {S 3 n } for n 8.
It was proved in [3] for large n and in [12] for all n that for k 4 the Turán number for P k 2 , or the maximum number of edges in a k-graph with no singleton intersection, is ex k (n; P k 2 ) = n−2 k−2 . In a couple of proofs we will need an easy analog of this result for k = 3, first observed in [12]. Fact 1. For n 1, we have ex 3 (n; P 3 2 ) n.

A hierarchy of Turán numbers
Turán numbers of the 1st order are just the ordinary Turán numbers defined above. Here we introduce a hierarchy of Turán numbers, where in each generation we consider only k-graphs which are not sub-k-graphs of extremal k-graphs from all previous generations. The next definition is iterative.      A historically first example of a Turán number of the 2nd order is due to Hilton and Milner [9] who determined the maximum size of a nontrivial intersecting family of k-sets, that is, one which is not a star. We state it here for k = 3 only and suppress the family Ex (2) 3 (n; M 3 2 ) which was also found in [9]. Set M := M 3 2 for convenience.
Theorem 5 ( [9]). For n 6, we have ex 3 (n; M ) = 3n − 8. In this paper we prove the following two results which we then use to compute some Ramsey numbers for P . First, we completely determine ex (2) 3 (n; P ), together with the corresponding 2-extremal 3-graphs. A comet Co(n) is a 3-graph with n vertices consisting of a copy of K 3 4 to which a star S 3 n−3 is attached, the unique common vertex being the center of the star (see Fig. 1). This vertex is called the center of the comet, while the set of the remaining three vertices of the 4-clique is called the head.
and Ex  and Ex  Note that for n 6 this number is not defined, since each 3-graph is a sub-3-graph of K 3 n . Then, we calculate the 3rd Turán number for P , but only for n = 12 which is, however, just enough for our application.

Conditional Turán numbers
To determine the Turán numbers of higher order, it is sometimes useful to rely on Theorem 5 and divide all 3-graphs into those which contain M and those which do not. This leads us quickly to another variation on Turán numbers.
Definition 3. For a family of k-graphs F, a family of F-free k-graphs G, and an integer n min{|V (G)| : G ∈ G}, the conditional Turán number is defined as ex k (n; F|G) = max{|E(H)| : |V (H)| = n, H is F-free, and ∃G ∈ G : H ⊇ G} Every n-vertex F-free k-graph with ex k (n; F|G) edges and such that H ⊇ G for some G ∈ G is called G-extremal for F. We denote by Ex k (n; F|G) the family of all n-vertex k-graphs which are G-extremal for F. (If F = {F } or G = {G}, we will simply write ex k (n; F |G), ex k (n; F|G), ex k (n; F |G), Ex k (n; F |G), Ex k (n; F|G), or Ex k (n; F |G), respectively.) In [11] we determined ex 3 (n; P |C) in terms of the ordinary Turán numbers ex 3 (n; P ).
Our next result reveals that the conditional Turán number ex(n; P |C) drops significantly if we restrict ourselves to connected 3-graphs only.
If H is a connected P -free 3-graph with n 7 vertices and H ⊃ C, then It is not a coincidence that in Lemma 1 and Theorem 5 we see the same extremal number 3n − 8. In fact, we prove Lemma 1 (see Section 5) by showing that the extremal 3-graph forms a nontrivial intersecting family.
Note that the Turán numbers ex 3 (n; P |M ) and ex (2) 3 (n; P ) coincide for n 8. We also find it useful to determine the Turán number for the pair {P, C} conditioning on 3-graphs H being non-intersecting.  Note that the Turán numbers ex 3 (n; {P, C}|M ), ex 3 (n; P |M ), and ex (2) 3 (n; P ) coincide for n 13.
To prove Theorem 10 we will need a lemma which states that if one, in addition to {P, C}, forbids also P 3 2 ∪ K 3 3 , then the formula, valid for ex 3 (n; {P, C}|M ) only for 6 n 9, takes over for all values of n.

Proof of Theorem 1
As mentioned in the Introduction, the inequality R(P ; r) r + 6, r 1, has been already proved in [10]. We are going to show that R(P ; r) r + 6 for each r = 4, 5, 6, 7.
Case r = 4. Let us consider an arbitrary 4-coloring of the 10 3 = 120 edges of the complete 3-graph K 3 10 . There exists a color with at least 1 4 · 120 = 30 edges. Denote the set of these edges by H. Since, by Theorem 4, Ex 3 10 }, and, by Theorem 6, ex (2) (10; P ) = 24 < 30, either P ⊆ H or H ⊆ S 3 10 . In the latter case we delete the center of the star containing H, together with the incident edges, obtaining a 3-coloring of K 3 9 . Since R(P ; 3) = 9, there is a monochromatic copy of P .
Case r = 5. The proof follows the lines of the previous one. We consider a 5-coloring of the complete 3-graph K 3 11 . There exists a color with at least 11 3 /5 = 33 edges. Denote the set of these edges by H. Again, by Theorems 4 and 6, either P ⊆ H or H ⊆ S 3 11 . In the latter case we delete the center of the star containing H, together with its incident edges, obtaining a 4-coloring of K 3 10 . Since, as we have just proved, R(P ; 4) = 10, there is a monochromatic copy of P .
Case r = 6. This is the most difficult case in which we have to appeal to the 3rd Turán number. We begin, as before, by considering an arbitrary 6-coloring of the complete 3-graph K 3 12 on the set of vertices V and assuming that it does not yield a monochromatic copy of the path P . Then none of the color classes can be contained in a star S 3 12 , since otherwise we would delete this star, obtaining a 5-coloring of K 3 11 , which surely contains a monochromatic P . By Theorems 4 and 6, S 3 12 and K 3 6 ∪ K 3 6 are, respectively, the unique 1-extremal and 2-extremal 3-graph for P . Consequently, by Theorem 7, every color class with more than 32 edges must be a sub-3-graph of K 3 6 ∪ K 3 6 . There exists a color class with at least 12 6 /6 = 37 edges which, as explained above, is contained in a copy K of K 3 6 ∪ K 3 6 . After deleting all the edges of K from K 3 12 , we obtain a complete bipartite 3-graph B with bipartition V = U ∪ W , |U | = |W | = 6, and with |E(B)| = 220 − 40 = 180 edges, colored by 5 colors. Note that any copy of K 3 6 ∪ K 3 6 may share with B at most 36 edges. Consequently, since 180/5 = 36, every color class has precisely 36 edges and, thus, is contained in K 3 6 ∪ K 3 6 . Let G i , i = 1, 2, 3, 4, 5, be the 5 color classes. Then, for each i, G i is fully characterized by two partitions, We now show that only 2 of the 5 color classes can be disjoint which is a contradiction (with a big cushion). For G 1 and G 2 to be disjoint, we need that {U 1 , U 1 } = {U 2 , U 2 } and {W 1 , W 1 } = {W 2 , W 2 }, which simply means that one of the partitions, of U or of W , must be swapped. But this implies that G 1 , G 2 , and G 3 cannot be pairwise disjoint.
4 Proofs of Theorems 6, 7, and 9 In this section we first deduce Theorems 6 and 7 from Lemma 1 and Theorems 9 and 10, with a little help of some already known results (Theorems 3-5). Then we deduce Theorem 9 from Corollary 1 and Theorem 10. The proofs of Lemmas 1 and 2 will be presented in the next section, while the proof of the crucial Theorem 10, based on Lemma 2, is deferred to the last section.
Throughout all the proofs, for convenience, we will be often identifying the edge set of a 3-graph with the 3-graph itself, writing, e.g., |H| instead of |E(H)|.
Otherwise, H is a {P, C}-free 3-graph containing M . Therefore, by Theorem 10, Proof of Theorem 9. Recall, that we want to determine the conditional Turán number ex 3 (n; P |M ). By considering whether or not a 3-graph contains a triangle, we infer that ex 3 (n; P |M ) = max{ex 3 (n; P |{M, C}), ex 3 (n; {P, C}|M )}. Theorem 9 follows now immediately from the respective parts of Corollary 1 and Theorem 10.

Proofs of Lemmas 1 and 2
For a 3-graph F and a vertex v ∈ V (F ) set F (v) = {e ∈ F : v ∈ e}. The degree of v in F is defined as |F (v)|.
Proof of Lemma 1.
Let H be a P -free, connected 3-graph with V (H) = V and |V | = n 7, containing a triangle. With some abuse of notation, we denote by C a fixed copy of the triangle in and let, recalling that we identify the edge set of a 3-graph with the 3-graph itself, Thus, the vertices x 1 , x 2 , x 3 are of degree two in C, while y 1 , y 2 , y 3 are of degree one. If H is an intersecting family (non-trivial due to the presence of C), then, by Theorem 5, |H| 3n − 8. We will show that if, on the other hand, H ⊇ M , then, in fact, |H| is even smaller. We begin with a simple observation.  If e ∈ T 1 , then one can easily check by inspection that C ∪ {e} ∪ {f } ⊃ P . Thus, e ∈ T 2 , say e ∩ U = {x 1 , x 2 }. The only edge in H[U ] disjoint from e which does not create a copy of the path P with C ∪ {e} is f = {x 3 , y 1 , y 2 } (see Fig. 4). Further, observe that all triples in T , except those of the type {x 1 , x 2 , w}, w ∈ W , form a copy of P with f and some edge of C.

Proof of Theorem 10
This section is entirely devoted to proving Theorem 10, that is, to determining the largest number of edges in an n-vertex 3-graph which is P -free and C-free but is not an intersecting family. First note that since |V (P 3 2 ∪ K 3 3 )| = 8, no n-vertex 3-graph, n = 6, 7, contains a copy of P 3 2 ∪ K 3 3 and therefore, by Lemma 2, Thus, from now on we will be assuming that n 8. Define a sequence of 3-graphs The main difficulty lies in showing the reverse inequality, namely, that any {P, C}-free 3-graph H on n 8 vertices, containing M , satisfies |H| |H n |. Moreover, for n 11, we want to show that the equality is reached by the extremal 3-graph H n = Co(n) only. We may assume that H contains a copy of P 3 2 ∪ K 3 3 , since otherwise, by Lemma 2, where the last inequality is strict for n 10. Before we turn to the actual proof of Theorem 10, we need to introduce some notation and prove preliminary results about the structure of H.

Preparations for the proof
We assume that H is {P, C}-free and contains a copy of P 3 2 ∪ K 3 3 . Let e 1 , e 2 ∈ H and x ∈ V = V (H) be such that e 1 ∩ e 2 = {x} and there is an edge in H disjoint from e 1 ∪ e 2 . We know that such a choice of e 1 , e 2 , x exists, because H ⊇ P 3 2 ∪ K 3 3 . We split V = U ∪ W , where U = e 1 ∪ e 2 , and W = V \ U.
We also split the set of edges of H(U, W ). First, notice that if for some h ∈ H(U, W ) we have |h ∩ U | = 1, then h ∩ U = {x}, since otherwise h together with e 1 and e 2 would form a copy of P in H. We let The edges h ∈ H(U, W ) with |h ∩ U | = 2 must satisfy h ∩ U ⊂ e 1 or h ∩ U ⊂ e 2 , since otherwise h together with e 1 and e 2 would form a copy of C in H. For k = 1, 2 define We have H(U, W ) = F 0 ∪ F 1 ∪ F 2 . (Note that in each case k = 0, 1, 2, the superscript k stands for the common size of the set h ∩ U \ {x} -see Fig. 7.) For a sub-3-graph F ⊆ H(U, W ) and i = 0, 1, set which in the important case of F = H(U, W ) will be abbreviated to H i . In particular, for i = 0, 1, , is the subset of edges of F k whose unique vertex in W lies in W i . A simple but crucial observation is that, since H is P -free, for every two disjoint edges in H, no edge may intersect each of them in exactly one vertex. Thus, there is no edge in H with one vertex in each of the sets, U , W 0 and W 1 . Therefore, and consequently, Furthermore, by the same principle, if e ∈ F 0 1 , then the pair e∩W 1 must be nonseparable in H[W 1 ], that is, every edge of H[W 1 ] must contain both these vertices or none. Since, as it can be easily proved, there are at most |W 1 | nonseparable pairs in W 1 , Another consequence of the above observation is that F 1 1 = ∅. Thus, To make use of (6), in addition to (5), we need to bound |F 2 1 | which, however, requires a detailed analysis of the degrees of vertices v ∈ W in the 3-graphs F k , k = 0, 1, 2. For v ∈ W and F ⊆ H, denote by F (v) the degree of v in F .
It can be easily checked that, since H is P -free, for every v ∈ W either Moreover, by the definitions of F 1 and F 2 , For v ∈ W 0 , by the remark preceding (3), |F 0 (v)| |W 0 | − 1, and thus, by (7), (8), and (1), In particular, for n = 10, while for n 11, where the equality for n 12 is achieved only when |F 0 (v)| = n − 9, |F 1 (v)| = 4, and For each e ∈ F 0 , the pair e ∩ W must be nonseparable and v belongs to at most two nonseparable pairs. Thus, |F 0 (v)| 2 and, consequently, by (7) and (8), One can also show, that and therefore, again by (8), Now we are ready to set bounds on the number of edges in H 1 , as well as in H[U ]∪H 1 , which will be repeatedly used in the proof of Theorem 10. Recall that |W 1 | 3.
Since H is C-free, on several occasions our proof relies on two instances of Theorem 3. Namely, if |W 0 | 1 then  Proof. Note that, due to P -freeness of H, the only edges allowed in H(U, V (Q)) with one vertex in U must belong to F 0 (there are at most two such edges). By symmetry, there are also at most two edges in H(U, V (Q)) with one vertex in W , which yields (18).

The proof
The structure of the proof is as follows. We first settle the three smallest cases, n = 8, 9, 10, one by one. Then we turn to the main case of n 11. Here, after quickly taking care of the easy subcase W 0 = ∅, we assume that W 0 = ∅ and proceed by induction on n with n = 11 being the base case. This part is a bit pedestrian, but afterwards, the induction step is almost immediate. Let H be a {P, C}-free n-vertex 3-graph which contains a copy of P 3 2 ∪K 3 3 . We adopt the notation and terminology from Subsection 6.1. In addition, for v ∈ V , we will write Moreover, by (9), |H(v)| 6, and consequently, |H| = |H − v| + |H(v)| 14 + 6 = 20. n 11. The proof is by induction on n with n = 11 being the base case. First, however, we take care of a simple subcase when W 0 = ∅, for which, by (14) and (17), Hence, in what follows we will be assuming that W 0 = ∅. In the remainder of this part of the proof, besides the assumption that W 0 = ∅, we will be also assuming that H[W ] is P 3 2 -free and thus, by Fact 1, |H[W ]| 6. We consider three cases with respect to the size of |W 0 |. |W 0 | = 1. We have |W 1 | = 5 and, by (13), |H 1 | 7. Consequently, by (4)  with equality only when |H 1 | = 3 and |H[U ∪ W 0 ]| = 21. The latter, by the second part of Theorem 3, is possible only when H[U ∪ W 0 ] is a star (with the center at x). This, in turn, implies that F 2 = ∅ (otherwise H would not be P -free) and, further, by (6), that H 1 = F 0 1 . Hence, H = Co (11) with x at the center and W 1 as the head.

Final comments
It would be interesting to decide if R(P ; r) = r + 6 for all r. If not, then what is the largest r 0 such that R(P ; r) = r + 6 for all r r 0 ? To even partially answer these questions, we would need to compute the conditional Turán numbers ex (s) (n; P |M ) for s 3.
For the related problem of computing R(C; r) it is only known that R(C; r) = r + 5 for r = 2, 3 and R(C; r) r + 5 for all r ( [7]). Gyarfas and Raeisi conjecture in [7] that R(C; r) = r + 5 for all r. To facilitate our approach to this problem one would need to compute ex (s) 3 (n; C) for s 2 and some small values of n. This would probably include calculating the conditional Turán numbers ex 3 (n; C|M ) = ex 3 (n; C|P ) which might be of independent interest. (The fact that the two numbers are the same was derived in [11] from Theorem 9 which was conjectured there.) In [11] we showed that ex 3 (n; C|M ) n−2 2 + 1 and conjectured that, indeed, this lower bound is the true value of ex 3 (n; C|M ).