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

IF 1 3区数学 Q1 MATHEMATICS

European Journal of Combinatorics Pub Date : 2024-04-12 DOI:10.1016/j.ejc.2024.103969

Deepak Bal , Louis DeBiasio , Allan Lo

引用次数: 0

Abstract

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 $Ω (r^{2} n) = {\hat{R}}_{r} (P_{n}) = O ((r^{2} log r) n)$ with the best bounds essentially due to Krivelevich (2019). In a recent breakthrough result, Letzter et al. (2021) gave a linear upper bound on the $r$ -color size-Ramsey number of the $k$ -uniform tight path $P_{n}^{(k)}$ ; i.e. ${\hat{R}}_{r} (P_{n}^{(k)}) = O_{r, k} (n)$ . At about the same time, Winter (2023) 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 ${\hat{R}}_{2} (P_{n}^{(k)}) \geq ({log}_{2} (k + 1)) n - O_{k} (1)$ 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 ${\hat{R}}_{r} (P_{n}^{(k)}) = Ω_{k} (r^{k} n)$ 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 ${\hat{R}}_{r} (P_{k + m}^{(k)}) = Θ_{k} (r^{m})$ for all $1 \leq m \leq k$ ; 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 $P_{n}^{(k, ℓ)}$ . In particular we note that when $1 \leq ℓ \leq \frac{k}{2}$ , we have $Ω_{k} (r^{2} n) = {\hat{R}}_{r} (P_{n}^{(k, ℓ)}) = O ((r^{2} log r) n)$ which essentially matches the best known bounds for graphs mentioned above. Additionally, in the case $k = 3$ , $ℓ = 2$ , and $r = 2$ , we give a more precise estimate which implies ${\hat{R}}_{2} (P_{n}^{(3)}) \geq \frac{28}{9} n - O (1)$ , improving on the above-mentioned lower bound of Winter in the case $k = 3$ .

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超图中路径的多色尺寸-拉姆齐数下限

k-uniform hypergraph H 的 r-color size-Ramsey number（用 Rˆr(H)表示）是 k-uniform hypergraph G 中最小的边数，对于 G 的边的每一个 r-coloring，都存在 H 的单色副本。在 2-uniform paths Pn 的情况下，已知 Ω(r2n)=Rˆr(Pn)=O((r2logr)n) ，最佳边界基本上是 Krivelevich（2019）提出的。在最近的一项突破性成果中，Letzter 等人（2021 年）给出了 k 条均匀紧密路径 Pn(k) 的 r 色大小-拉姆齐数的线性上界，即 Rˆr(Pn(k))=Or,k(n)。大约与此同时，Winter (2023) 首次给出了 k≥3 时 Pn(k) 的双色大小-拉姆齐数的非难下界；即 Rˆ2(Pn(3))≥83n-O(1) 和 k≥4 时 Rˆ2(Pn(k))≥log2(k+1)n-Ok(1) 。我们的主要结果是 Rˆr(Pn(k))=Ωk(rkn)，它概括了上述已知图形下限。我们证明的关键要素之一是一个有趣的结果。我们证明，在所有 1≤m≤k 的情况下，Rˆr(Pk+m(k))=Θk(rm)；也就是说，我们确定了每条足够短的紧密路径的 r 色大小-拉姆齐数的正确数量级。我们的所有结果都可以推广到 ℓ-overlapping k-uniform paths Pn(k,ℓ)。我们特别注意到，当 1≤ℓ≤k2 时，我们有 Ωk(r2n)=Rˆr(Pn(k,ℓ))=O((r2logr)n)，这基本上与上述图的已知最佳边界相吻合。此外，在 k=3、ℓ=2 和 r=2 的情况下，我们给出了一个更精确的估计值，即 Rˆ2(Pn(3))≥289n-O(1)，比上述 k=3 情况下的 Winter 下限有所提高。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

European Journal of Combinatorics 数学-数学

CiteScore

2.10

自引率

10.00%

发文量

124

审稿时长

4-8 weeks

期刊介绍： The European Journal of Combinatorics is a high standard, international, bimonthly journal of pure mathematics, specializing in theories arising from combinatorial problems. The journal is primarily open to papers dealing with mathematical structures within combinatorics and/or establishing direct links between combinatorics and other branches of mathematics and the theories of computing. The journal includes full-length research papers on important topics.