A lower bound on the multicolor size-Ramsey numbers of paths in hypergraphs

The $r$-color size-Ramsey number of a $k$-uniform hypergraph $H$, denoted by $\hat{R}_r(H)$, is the minimum number of edges in a $k$-uniform hypergraph $G$ such that for every $r$-coloring of the edges of $G$ there exists a monochromatic copy of $H$. In the case of $2$-uniform paths $P_n$, it is known that $\Omega(r^2n)=\hat{R}_r(P_n)=O((r^2\log r)n)$ with the best bounds essentially due to Krivelevich. In a recent breakthrough result, Letzter, Pokrovskiy, and Yepremyan gave a linear upper bound on the $r$-color size-Ramsey number of the $k$-uniform tight path $P_{n}^{(k)}$; i.e. $\hat{R}_r(P_{n}^{(k)})=O_{r,k}(n)$. Winter gave the first non-trivial lower bounds on the 2-color size-Ramsey number of $P_{n}^{(k)}$ for $k\geq 3$; i.e. $\hat{R}_2(P_{n}^{(3)})\geq \frac{8}{3}n-O(1)$ and $\hat{R}_2(P_{n}^{(k)})\geq \lceil\log_2(k+1)\rceil n-O_k(1)$ for $k\geq 4$. We consider the problem of giving a lower bound on the $r$-color size-Ramsey number of $P_{n}^{(k)}$ (for fixed $k$ and growing $r$). Our main result is that $\hat{R}_r(P_n^{(k)})=\Omega_k(r^kn)$ which generalizes the best known lower bound for graphs mentioned above. One of the key elements of our proof is a determination of the correct order of magnitude of the $r$-color size-Ramsey number of every sufficiently short tight path; i.e. $\hat{R}_r(P_{k+m}^{(k)})=\Theta_k(r^m)$ for all $1\leq m\leq k$. All of our results generalize to $\ell$-overlapping $k$-uniform paths $P_{n}^{(k, \ell)}$. In particular we note that when $1\leq \ell\leq \frac{k}{2}$, we have $\Omega_k(r^{2}n)=\hat{R}_r(P_{n}^{(k, \ell)})=O((r^2\log r)n)$ which essentially matches the best known bounds for graphs mentioned above. Additionally, in the case $k=3$, $\ell=2$, and $r=2$, we give a more precise estimate which implies $\hat{R}_2(P^{(3)}_{n})\geq \frac{28}{9}n-O(1)$, improving on the above-mentioned lower bound of Winter in the case $k=3$.


(k)
n ; i.e.Rr (P At about the same time, Winter [36] gave the first non-trivial lower bounds on the 2-color size-Ramsey number of P (k) n for k ≥ 3; i.e.R2 (P n ) ≥ 8  3 n − O(1) and R2 (P We consider the problem of giving a lower bound on the r-color size-Ramsey number of P (k) n (for fixed k and growing r).Our main result is that Rr (P (k) n ) = Ω k (r k n) which generalizes the best known lower bound for graphs mentioned above.One of the key elements of our proof turns out to be an interesting result of its own.We prove that Rr (P (k) k+m ) = Θ k (r m ) for all 1 ≤ m ≤ k; that is, we determine the correct order of magnitude of the r-color size-Ramsey number of every sufficiently short tight path.
All of our results generalize to ℓ-overlapping k-uniform paths P (k,ℓ) n . In particular we note that when 1 ≤ ℓ ≤ k 2 , we have Ω k (r 2 n) = Rr (P

Introduction
Given hypergraphs G and H and a positive integer r, we write G → r H to mean that in every r-coloring of the edges of G, there exists a monochromatic copy of H. Given a k-uniform hypergraph H, the r-color size-Ramsey number of H, denoted by Rr (H), is the minimum number of edges in a k-uniform hypergraph G such that G → r H.The r-color Ramsey number of H, denoted by R r (H), is the minimum number of vertices in a k-uniform hypergraph G such that G → r H.If r = 2, then we drop the subscript.For all integers 0 ≤ ℓ ≤ k−1 and positive integer m, a k-uniform ℓ-overlapping path (or a (k, ℓ)-path for short) with m edges is a k-uniform graph on vertex set {v 1 , . . ., v k+(m−1)(k−ℓ) } with edges {v (i−1)(k−ℓ)+1 , . . ., v (i−1)(k−ℓ)+k } for all i ∈ [m] (note that the case ℓ = 0 corresponds to a matching).Note that any pair of consecutive edges has exactly ℓ vertices in common.A (k, ℓ)-path with n vertices, denoted by P (k,ℓ) n , has n−ℓ k−ℓ edges (so whenever we write P (k,ℓ) n , we are implicitly assuming that k − ℓ divides n − ℓ).We sometimes write P (k,ℓ) ℓ+m(k−ℓ) to emphasize that the (k, ℓ)-path has m edges.If ℓ = 0, 1, k − 1, then P (k,ℓ) n is a k-uniform matching, a k-uniform loose path and a k-uniform tight path, respectively.We write P (k) n and P n for P (k,k−1) n and P (2) n , respectively.For graphs, Beck [6] proved R(P n ) = O(n) and the best known bounds [5,8] are (3.75 − o(1))n ≤ R(P n ) < 74n.For the r-color version, it is known [5,9,10,27] that Ω(r 2 n) = Rr (P n ) = O((r 2 log r)n). (1.1) After a number of partial results, e.g.[7,20,31], Letzter, Pokrovskiy, and Yepremyan [29] recently proved that for all r ≥ 2 and 1 ≤ ℓ ≤ k − 1, Rr (P (their result covers more than just the case of paths, but for simplicity we don't state their general result here).At around the same time, Winter provided the first non-trivial lower bounds on the size-Ramsey number of P (k,ℓ) n for 2 colors.Theorem 1.1 (Winter [35,36]).For all integers n ≥ k ≥ 2, (i) R(P In this paper, we prove a general lower bound on the r-color size-Ramsey number of P (k,ℓ) n . Our lower bound depends on both the r-color Ramsey number of the path P (k,ℓ) n (for which the order of magnitude R r (P and the r-color size-Ramsey number of the "short" path P (for which essentially nothing was known).See the first paragraph of Section 3 for an explanation of the significance of P Rr (P ).For all integers n ≥ c 0 , we have In particular, if ℓ = k − 1, then Rr (P As mentioned above, it is known that R r (P More specifically, for all integers r, k ≥ 2 and sufficiently large n, where the upper bound follows from a result of Allen, Böttcher, Cooley, Mycroft [1] and the lower bound is due to Proposition 2.1 (there are more precise results for some specific values of k and r, but we defer this discussion to Section 2).As a result, in order to get an explicit lower bound from Theorem 1.2, it remains to get a lower bound on Rr (P ), where In fact, we are able to determine Rr (P ) exactly up to a constant factor depending on k; more specifically, we show Rr (P (or in the case of tight paths, Rr (P ).We obtain this result as a consequence of the following more general theorem.
In particular, when ℓ = k − 1, this says that for all 1 ≤ m ≤ k, Rr (P ).In Section 6, we have a further discussion about the hidden constants in Theorem 1.3.From (1.2) and Theorems 1.2 and 1.3 (with the appropriate reindexing -see the proof in Section 5), we get the following corollary.
(1.3)So as a consequence of Theorem 1.5 and (1.3), we get a refinement of Corollary 1.4 for tight paths when k = 3.

Notation
We sometimes write k-graph to mean k-uniform hypergraph and k-path to mean k-uniform tight path.Given a k-graph H and a vertex v ∈ V (H), the link graph of v in H, denoted by H v , is the (k − 1)-graph on V (H) such that e ∈ V (H) k−1 is an edge of H v if and only if e ∪ {v} is an edge of H.For a subset S ⊆ V (H), we write H \ S to be the sub-k-graph of H obtained by deleting vertices of S. We sometimes also identify H with its edge set.
We write f (r, k, d) = O k,d (g(r, k, d)) to mean that there exists a constant C k,d , possibly depending on k and d, such that f For integers a and b, we write [b] for {1, . . ., b} and [a, b] for {a, a + 1, . . ., b}.

Organization of Paper
In Section 2, we give lower bounds on the Ramsey numbers of (k, ℓ)-paths and upper bounds on the size Ramsey numbers of short tight paths.In Section 3 we prove Theorem 1.2, and in Section 4 we prove Theorem 1.3.Section 5 contains proofs of the remaining results discussed above.Finally, we discuss some open problems in Section 6.
2 Relationship between size-Ramsey numbers, Ramsey numbers, and Turán numbers of paths While there are many papers which give exact results [15,17], asymptotic results [18,22,23,30], and bounds [26] on R r (P ) for some particular values of r, k, ℓ, the general problem of (asymptotically) determining R r (P ) is completely wide open.For our purposes, we will only need a lower bound on R(P (k,ℓ) n ) which holds for all n, and an upper bound on R(P (k) n ) which holds for n = O(k).We note that even if the exact value of R r (P (k) n ) was known for all r, k, n ≥ 2, the only effect it would have on our results is slightly improving the hidden constants in Corollary 1.4.

A lower bound on the Ramsey number of (k, ℓ)-paths
Recall that P (k,0) km is the k-uniform matching of size m.The following proposition gives a lower bound on R r (P ), which is based on the construction of Alon, Frankl, and Lovász [3] giving a lower bound on R r (P (k,0) km ), and was essentially already observed in [23].
In particular, when k − ℓ divides k, we have R r (P contains a matching of size m ′ .
Let A 1 , . . ., A r−1 be disjoint vertex sets each of order m ′ − 1 and one vertex set A r of order n − 1.Then take a complete k-graph on A 1 ∪ • • • ∪ A r and color each edge e with the smallest index i such that e ∩ A i = ∅.Note that for all i ∈ [r − 1], there is no matching of color i of order m ′ and thus there is no (k, ℓ)-path of color i of order n.Also, all edges of color r are contained in A r which is of order n − 1, so there is no (k, ℓ)-path of color r of order n.So we have R r (P We also note that in the case of graphs it is known [37] that R r (P n ) ≥ (r − 1)(n − 1) for all r ≥ 3, which is better than the bound above when k = 2 and r ≥ 4.
We now consider the case when r is sufficiently large compared to k and n, in which case we can improve the lower bound by a factor of k for infinitely many r.Let K N exists for sufficiently large N subject to some necessary divisibility conditions.Glock, Kühn, Lo, and Osthus [16] gave an alternative proof1 .Theorem 2.2 (Keevash [25], Glock, Kühn, Lo, and Osthus [16]).For all integers 1 ≤ k ≤ n, there exists We use Theorem 2.2 to extend a result of Axenovich, Gyárfás, Liu, and Mubayi [4, Theorem 11] who proved that R r (P 1)) for all r ≥ 2. As in [4], we use the following result of Pippenger and Spencer [34] (stated here only for regular hypergraphs).
Theorem 2.3 (Pippenger and Spencer [34]).Every D-regular, m-uniform hypergraph having the property that every pair of vertices is contained together in o(D) edges can be decomposed into (1 + o(1))D many matchings.Proposition 2.4.For all integers n > k ≥ 2 there exists an r 0 = r 0 (n, k) such that for infinitely many integers r ≥ r 0 , R r (P n−1 containing S ∪ S ′ .Hence every pair of vertices in H is contained together in at most one edge.Now Theorem 2.3 implies that H can be decomposed into at most (1 + o(1))(N − k + 1)/(n − k) matchings, and we color each such matching with a distinct color.Let r be the number of colors used, so n−1 which pairwise intersect in at most k − 2 vertices, so the largest tight component has order n − 1 and thus there are no monochromatic copies of P (k) n .

An upper bound on the size-Ramsey number of short tight paths
We will later need an upper bound on Rr (P ; however, we will prove in Lemma 4.2 that Rr (P 2m−2 ).As a result, we only discuss tight paths in this subsection.
For graphs, it is known that which is equivalent to determining the edge chromatic number of complete graphs.Bierbrauer [6] proved the following surprisingly difficult result Given a k-graph H, let ex(n, H) be the maximum number of edges in a k-graph G on n vertices such that H ⊆ G.Note that if G is a k-graph on n vertices with more than ex(n, H) edges, then H ⊆ G.
Kalai conjectured (see [11]) that for all integers N ≥ n ≥ k ≥ 2, ex(N, P was given explicitly by Györi, Katona, and Lemons [19,Theorem 1.11], and in more generality by Füredi and Jiang [12] (see [13] for further discussion).The best known bounds are due to Füredi, Jiang, Kostochka, Mubayi, and Verstraëte [14], but as their result depends on the parity of k and only improves on (2.2) by about a factor of 2, we will use the simpler bound given in (2.2).We will also use the following trivial observation Combining (2.2) and (2.3) we have the following corollary (which will be useful when n = O(k)).
Corollary 2.5.For all integers n, r, k ≥ 2, R(P n ) ≤ rkn and thus Rr (P Proof.We first show that R(P so the majority color class in any r-coloring of rkn contains a copy of n by (2.2).Thus by (2.3), we have Rr (P Note that if Kalai's conjecture is true, then we would have Rr (P 3 Proof of Theorem 1.2 In this section we prove Theorem 1.2 (which can be thought of as a generalization of [5, Lemma 2.4, Corollary 2.5]).Given a (k, ℓ)-path P = v 1 v 2 . . .v p , let the exterior of P , denoted by P ext , be the set consisting of the first ℓ vertices and the last ℓ vertices of P ; that is P ext = {v 1 , . . ., v ℓ } ∪ {v p−ℓ+1 , . . ., v p }, and let the interior of P , denoted by P int , be V (P ) \ P ext (so for example in the case k = 2, the endpoints of the path form the exterior and the remaining vertices form the interior).The key observation is that if v ∈ P int , then the link graph of v in P is either a (k −1, ℓ−1)-path with k k−ℓ edges or a (k −1, ℓ−1)-path with k k−ℓ edges (and this is not necessarily the case if v ∈ P ext ).
Proof of Theorem 1.2.Recall that , and We will show that there is an r-coloring of H with no monochromatic n ′ ) and thus we can color H \ S with r colors so there is no monochromatic Let E S be the set of edges from E which intersect S and let H S be the k-graph induced by edges in E S .Note that for every vertex in the interior of a (k, ℓ)-path, its link graph contains a (k − 1, ℓ − 1)-path with k k−ℓ edges; i.e.P .Recall that every v ∈ S has degree less than d = Rr (P ) in H S .So if we can color the edges of H S so that there is no monochromatic P (k−1,ℓ−1) q in the link graph of every v ∈ S, then no vertex of S will be in the interior of any monochromatic (k, ℓ)-paths.Thus H would have no monochromatic (k, ℓ)-path of order n ′ + 2ℓ (as there are no monochromatic (k, ℓ)-paths of order n ′ in H − S and there are 2ℓ vertices in the exterior of a (k, ℓ)-path).However, we cannot color each link graph of v ∈ S independently since it is possible for an edge to contain more than one vertex from S. Instead, we are able to provide an r-coloring having the property that if a vertex of S is in the interior of a monochromatic (k, ℓ)-path, then it must be within distance k 2 d from an endpoint of P (which explains the reason for the definition of n ′ ).
We begin by partitioning S into sets S 1 , . . ., S c such that for all e ∈ E S and all i ∈ [c], |e ∩ S i | ≤ 1 with c as small as possible.Since H S is a k-uniform hypergraph with maximum degree at most d − 1, the usual greedy coloring algorithm (color the vertices one by one and note that each vertex is contained in an edge with at most (k − 1)( d − 1) vertices which have already been colored) implies that For all i ∈ [c], let E i be the set of edges which are incident with a vertex in S i but no vertices in S 1 ∪ • • • ∪ S i−1 and let H i be the k-graph induced by edges in E i .Note that {E 1 , . . ., E c } is a partition of E S .For each i ∈ [c], E i can be further partitioned based on which vertex in S i they are adjacent to.For all i ∈ [c] and all v ∈ S i , we color the edges of E i incident with v so that there are no monochromatic copies of P (k−1,ℓ−1) q in the link graph of v in H i (which is possible by the definition of d and the previous sentences).
We now show that H does not contain a monochromatic (k, ℓ)-path on n vertices.
Proof of Claim 3.1.Let j 0 = j and i 0 ∈ [c] be such that v j 0 ∈ S i 0 .If v j 0 ∈ P ext , then min{j, p − j + 1} ≤ ℓ and so we are done.If v j 0 ∈ P int , then we will show that there is a vertex v j 1 ∈ S i 1 such that |j 1 − j 0 | ≤ k − 1 and i 1 < i 0 .We then repeat this argument for v j 1 and note that this will stop within c rounds.
Suppose that we have found j a and i a for some a ≥ 0. If that is, v ja ∈ P int .Note that the link graph of v ja in P contains P (k−1,ℓ−1) q .Thus by the coloring of the edges in E S , there must exist j a+1 ∈ [j a − k + 1, j a + k − 1] and i a+1 < i a such that v j a+1 ∈ S i a+1 .
By Claim 3.1, we know that if we remove ℓ + c(k − 1) vertices from each end of P , then the resulting (k, ℓ)-path lies in H \S, which has order at most n ′ −1 by the coloring of H \S. Therefore, we have 4 Size-Ramsey numbers of short (k, ℓ)-paths
(i) For all 1 ≤ ℓ ≤ k 2 , Rr (P ). (ii) For all n ≥ k > ℓ ≥ 1 and d ≥ 1 such that d divides n, k, and ℓ, Rr (P Proof.For (i), let m = n−ℓ k−ℓ , so that P The following lemma provides an upper bound on the r-color size-Ramsey number of every sufficiently short (k, ℓ)-path.
In particular (when m = 1), we have Rr (P ) edges such that H → r Rr (P ) and create a k-graph H ′ having the same number of edges as H by adding a set U of k − m(k − ℓ) vertices which are contained in every edge of H. Now in every r-coloring of the edges of H ′ (and by extension H), there is a monochromatic copy of P in H and thus a monochromatic copy of P

Lower bound
The following lemma will be applied with r−1 2f (k,ℓ,m) in place of r to give us a lower bound on Rr (P . Before beginning the proof, we give a high level overview.We are given a k-uniform hypergraph H with a sufficiently small number of edges which we must color in such a way that there is no monochromatic P (k,ℓ) ℓ+(m+1)(k−ℓ) .In order to produce such a coloring, we actually prove something stronger.We say that a pair of edges (e, f ) ∈ E(H) × E(H) is "dangerous" if |e ∩ f | = ℓ and they satisfy an additional special property (see (4.1) below).As we will prove, every copy of P (k,ℓ) ℓ+(m+1)(k−ℓ) contains a dangerous pair of edges.Furthermore, the definition of dangerous will imply that for every edge e ∈ E(H), there are only a bounded number of f ∈ E(H) for which (e, f ) is a dangerous pair.This allows us to greedily color the edges of H so that no dangerous pair of edges receives the same color and therefore there is no monochromatic copy of P (k,ℓ) ℓ+(m+1)(k−ℓ) .Now we proceed to the formal proof.
Proof of Lemma 4.3.Set B 1 = H.For all j ∈ [2, m + 1], let B j be the t j -graph on V (H) such that In words, for all j ∈ [2, m + 1], B j is a t j -graph, whose edges are those with degree "too large" in B j−1 .We first establish the following claim.Note that when m = k k−ℓ and k − ℓ divides k, B m+1 is the 0-graph and we take B m+1 = ∅ to mean that the empty set is not an edge of B m+1 ; that is, B m has at most r edges.
Proof of claim.If B m+1 = ∅, then there is some t m+1 -set which has degree greater than r in B m and thus |B m | > r.Observe that for all j ∈ [2, m], Hence We begin by showing that there exist i * , j * ∈ [m] such that e i * ∩ e i * +j * −1 , e i * +1 ∩ e i * +j * ∈ B j * and e i * ∩ e i * +j * / ∈ B j * +1 .For all j ∈ [m + 1], let Q j be the (t j , t j+1 )-path with consecutive edges e 1 ∩ e j , . . ., e m+2−j ∩ e m+1 .Note that Q 1 ⊆ H = B 1 and Q m+1 ⊆ B m+1 by Claim 4.4 (Again, if m = k k−ℓ and k −ℓ divides k, then Q m+1 contains the empty set as an edge but B m+1 does not.).Thus there exists a smallest j We now show that (e i * , e i * +1 ) is dangerous.Let S = e i * ∩ e i * +j * ⊆ e i * ∩ e i * +1 , so S ∈ e i * ∩e i * +1 Hence S ∪ (e i * +1 \ e i * ) = e i * +1 ∩ e i * +j * ∈ B j * .Therefore, (e i * , e i * +1 ) is dangerous.
We next show that for every e ∈ E(H), there are a bounded number of e ′ ∈ E(H) such that (e, e ′ ) is dangerous.
where the last inequality follows from Claim 4.6.
We now prove Theorem 1.3.
5 Corollaries, more precise bounds, and a further extension To give an improved lower bound on R(P 4 ), we first need a few definitions.The arboricity of a graph G, denoted by arb(G), is the smallest number of forests needed to decompose the edge set of G.The star arboricity of a graph G, denoted by arb ⋆ (G), is the smallest number of star-forests needed to decompose the edge set of G.Note that since every forest can be decomposed into at most two star-forests we have that for all G, arb ⋆ (G) ≤ 2arb(G).A well-known result of Nash-Williams [32,33]   , where the last inequality holds by (1.3) and Theorem 1.5 respectively.
(k,ℓ) n has m edges.Begin with a graph G such that G → r P m+1 .Now replace each vertex with a set of ℓ vertices and add an additional k − 2ℓ unique vertices to each edge to get a k-graph H with the same number of edges as G such that H → r P (k,ℓ) n (any r-coloring of H corresponds to an r-coloring of G, and any monochromatic P m+1 in G corresponds to a monochromatic P(k,ℓ) n in H).For (ii), begin with a (k/d)-graph G such that G → r P (k/d,ℓ/d) n/d.Now replace each vertex with a set of d vertices to get a k-graph H with the same number of edges as G such that H → r P (k,ℓ) n .For (iii), note that P (k,ℓ) ℓ+(m+1)(k−ℓ) contains a vertex v such that its link graph is precisely P (k−1,ℓ−1) ℓ−1+(m+1)(k−ℓ) .Begin with a (k − 1)-graph G such that G → r P (k−1,ℓ−1) ℓ−1+(m+1)(k−ℓ) .Now let y be a new vertex and add y to each edge of G to get a k-graph H with the same number of edges as G such that H → r P (k,ℓ)

Claim 4 . 6 .
For all e ∈ E(H), there are at most r • f (k, ℓ, m) many e ′ ∈ E(H) such that (e, e ′ ) is dangerous.Proof of claim.Let e ∈ E(H).For all j ∈ [m], we count the number of e ′ ∈ E(H) such that (e, e ′ ) satisfy (4.1) with j * = j.There are at most k ℓ choices for e ∩ e ′ and at most ℓ t j+1 choices for S ∈ e∩e ′ t j+1 \ B j+1 .Since S / ∈ B j+1 and S ∪ (e ′ \ e) ∈ B j , there are at most r choices for e ′ \ e. Hence there are at most r k ℓ ℓ t j+1 many e ′ ∈ E(H) such that (e, e ′ ) satisfy (4.1) with j * = j.Therefore the total number of e ′ ∈ E(H) such that (e, e ′ ) satisfy (4.1) is at most r • f (k, ℓ, m) as required.Finally, we use Claim 4.6 to color the edges of H such that no dangerous pair receives the same color.Since every copy of P (k,ℓ) ℓ+(m+1)(k−ℓ) contains a dangerous pair of edges by Claim 4.5, this implies that such a coloring of H does not contain a monochromatic P (k,ℓ) ℓ+(m+1)(k−ℓ) .Let D be the auxiliary digraph such that V (D) = E(H) and (e, e ′ ) ∈ E(D) if and only if (e, e ′ ) is dangerous.Let G be the underlying undirected graph of D. A proper vertex-coloring of G gives an edge-coloring of H (with the same set of colors) without a monochromatic P (k,ℓ)

First we prove Corollary 1 . 4 ,
then we obtain the more precise bounds given in Theorem 1.5, Corollary 1.6, and Theorem 1.7.At the end, we mention one more extension of our results.Proof of Corollary 1.4.By Theorem 1.3