拉姆齐理论的一个核心问题是，$N$顶点上的图中最大团和独立集的大小可以有多小。根据Erdés在1947年的著名结果，边概率为$1/2$的$N$顶点上的随机图，以高概率不包含大于$2\log_2 N$的团或独立集。寻找具有类似Ramsey型性质的图的显式结构是一个著名的公开问题。一种自然的方法是使用代数工具来构造这样的图。称$r$ -一致超图$\mathcal{H}$是\emph{algebraic of complexity $(n,d,m)$}，如果$\mathcal{H}$的顶点是某个域$\mathbb{F}$的$\mathbb{F}^{n}$的元素，并且存在次数至多为$d$的$m$多项式$f_1,\dots,f_m:(\mathbb{F}^{n})^{r}\rightarrow \mathbb{F}$，使得$\mathcal{H}$的边由$f_1,\dots,f_m$的零模式决定.本文的目的是证明，如果一个代数图（或超图）的复杂性$(n,d,m)$具有良好的Ramsey性质，则至少有一个参数$n,d,m$必须是大的.在2001年，Rónyai，Babai和Ganapathy考虑了Ramsey问题的二部变体，并证明了如果$G$是一个在$N$顶点上的复杂度为$(n,d,m)$的代数图，则$G$或其补图包含一个大小为$Ω_{n,d,m}(N^{1/(n+1)})$的完全平衡二部图。我们扩展这一结果表明，这样的$G$包含一个集团或一个独立的一套大小$N^{Ω(1/ndm)}$和证明类似的结果代数超图的恒定复杂性。我们还得到了一个多项式正则性引理$r$ -一致代数超图是由一个单一的多项式，这可能是独立的利益。我们的证明结合了代数，几何和组合工具。
英文摘要：
One of the central questions in Ramsey theory asks how small can be the size of the largest clique and independent set in a graph on $N$ vertices. By the celebrated result of Erdős from 1947, the random graph on $N$ vertices with edge probability $1/2$, contains no clique or independent set larger than $2\log_2 N$, with high probability. Finding explicit constructions of graphs with similar Ramsey-type properties is a famous open problem. A natural approach is to construct such graphs using algebraic tools. Say that an $r$-uniform hypergraph $\mathcal{H}$ is \emph{algebraic of complexity $(n,d,m)$} if the vertices of $\mathcal{H}$ are elements of $\mathbb{F}^{n}$ for some field $\mathbb{F}$, and there exist $m$ polynomials $f_1,\dots,f_m:(\mathbb{F}^{n})^{r}\rightarrow \mathbb{F}$ of degree at most $d$ such that the edges of $\mathcal{H}$ are determined by the zero-patterns of $f_1,\dots,f_m$. The aim of this paper is to show that if an algebraic graph (or hypergraph) of complexity $(n,d,m)$ has good Ramsey properties, then at least one of the parameters $n,d,m$ must be large. In 2001, Rónyai, Babai and Ganapathy considered the bipartite variant of the Ramsey problem and proved that if $G$ is an algebraic graph of complexity $(n,d,m)$ on $N$ vertices, then either $G$ or its complement contains a complete balanced bipartite graph of size $Ω_{n,d,m}(N^{1/(n+1)})$. We extend this result by showing that such $G$ contains either a clique or an independent set of size $N^{Ω(1/ndm)}$ and prove similar results for algebraic hypergraphs of constant complexity. We also obtain a polynomial regularity lemma for $r$-uniform algebraic hypergraphs that are defined by a single polynomial, that might be of independent interest. Our proofs combine algebraic, geometric and combinatorial tools.
Much recent progress in hypergraph Ramsey theory has focused on constructions that lead to lower bounds for the corresponding Ramsey numbers. In this paper, we consider applications of these results to Gallai colorings. That is, we focus on the Ramsey numbers resulting from only considering $t$-colorings of the hyperedges of complete $r$-uniform hypergraphs in which no rainbow $K_{r+1}^{(r)}$-subhypergraphs exists. We also provide new constructions which imply improved lower bounds for many $3$ and $4$-uniform Ramsey numbers and $3$ and $4$-uniform Gallai-Ramsey numbers.
Let $K_m^{(3)}$ denote the complete $3$-uniform hypergraph on $m$ vertices and $S_n^{(3)}$ the $3$-uniform hypergraph on $n+1$ vertices consisting of all $\binom{n}{2}$ edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off-diagonal Ramsey number $r(K_{4}^{(3)},S_n^{(3)})$ exhibits an unusual intermediate growth rate, namely, \[ 2^{c \log^2 n} \le r(K_{4}^{(3)},S_n^{(3)}) \le 2^{c' n^{2/3}\log n} \] for some positive constants $c$ and $c'$. The proof of these bounds brings in a novel Ramsey probl…
An edge-colored hypergraph is rainbow if all of its edges have different colors. Given two hypergraphs $\mathcal{H}$ and $\mathcal{G}$, the anti-Ramsey number $ar(\mathcal{G}, \mathcal{H})$ of $\mathcal{H}$ in $\mathcal{G}$ is the maximum number of colors needed to color the edges of $\mathcal{G}$ so that there does not exist a rainbow copy of $\mathcal{H}$. Li et al. determined the anti-Ramsey number of $k$-matchings in complete bipartite graphs. Jin and Zang showed the uniqueness of the extremal coloring. In this paper, as a generalization of these results, we determine the anti-Ramsey numbe…
We give asymptotically optimal constructions in generalized Ramsey theory using results about conflict-free hypergraph matchings. For example, we present an edge-coloring of $K_{n,n}$ with $2n/3 + o(n)$ colors such that each $4$-cycle receives at least three colors on its edges. This answers a question of Axenovich, Füredi and the second author (On generalized Ramsey theory: the bipartite case, J. Combin. Theory Ser B 79 (2000), 66--86). We also exhibit an edge-coloring of $K_n$ with $5n/6+o(n)$ colors that assigns each copy of $K_4$ at least five colors. This gives an alternative very short s…
For an integer $k \geq 2$, an ordered $k$-uniform hypergraph $\mathcal{H}=(H,<)$ is a $k$-uniform hypergraph $H$ together with a fixed linear ordering $<$ of its vertex set. The ordered Ramsey number $\overline{R}(\mathcal{H},\mathcal{G})$ of two ordered $k$-uniform hypergraphs $\mathcal{H}$ and $\mathcal{G}$ is the smallest $N \in \mathbb{N}$ such that every red-blue coloring of the hyperedges of the ordered complete $k$-uniform hypergraph $\mathcal{K}^{(k)}_N$ on $N$ vertices contains a blue copy of $\mathcal{H}$ or a red copy of $\mathcal{G}$. The ordered Ramsey numbers are quite extensivel…
给定一个超图$H$，大小Ramsey数$\hat{r}_2(H)$是最小整数$m$，使得存在一个具有$m$边的图$G$，且该图的性质是：在$G$的边的任意着色中，都有$H$的单色副本.证明了$n$顶点$P^{(3)}_n$上的$3$ -一致紧路的大小Ramsey数在$n$中是线性的，即，$\hat{r}_2(P^{(3)}_n) = O(n)$。这回答了Dudek，Fleur，Mubayi和Rödl关于$3$ -一致超图的问题[On the size-Ramsey number of hypergraphs，J.图论86（2016），417-434]，他们证明了$\hat{r}_2(P^{(3)}_n) = O(n^{3/2} \log^{3/2} n)$。
