3-Uniform hypergraphs of bounded degree have linear Ramsey numbers

递归上界与顶点在线Ramsey游戏的递归上界及其在超图Ramsey数中的应用

Recursive upper bounds for the vertex online Ramsey game with applications to hypergraph Ramsey numbers

本文研究了超图Ramsey数的递归上界，提出改进的递推关系并应用于超图Ramsey数的上界优化。

A note on lower bounds for hypergraph Ramsey numbers

We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \[r_3 (l,l,l) \geq 2^{l^{c \log \log l}}.\] The old bound, due to Erdős and Hajnal, was \[r_3 (l,l,l) \geq 2^{c l^2 \log^2 l}.\]

A survey of hypergraph Ramsey problems

The classical hypergraph Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple among them is blue. We survey a variety of problems and results in hypergraph Ramsey theory that have grown out of understanding the quantitative aspects of $r_k(s,n)$. Our focus is on recent developments and open problems.

Embeddings and Ramsey numbers of sparse k-uniform hypergraphs

Chvatal, Roedl, Szemeredi and Trotter proved that the Ramsey numbers of graphs of bounded maximum degree are linear in their order. In previous work, we proved the same result for 3-uniform hypergraphs. Here we extend this result to k-uniform hypergraphs, for any integer k > 3. As in the 3-uniform case, the main new tool which we prove and use is an embedding lemma for k-uniform hypergraphs of bounded maximum degree into suitable k-uniform `quasi-random' hypergraphs.

A class of Ramsey-extremal hypergraphs

In 1991, McKay and Radziszowski proved that, however each 3-subset of a 13-set is assigned one of two colours, there is some 4-subset whose four 3-subsets have the same colour. More than 25 years later, this remains the only non-trivial classical Ramsey number known for hypergraphs. In this article, we find all the extremal colourings of the 3-subsets of a 12-set and list some of their properties. Using the catalogue, we provide an answer to a question of Dudek, Fleur, Mubayi and Rödl about the size-Ramsey numbers of hypergraphs.

Size-Ramsey numbers of powers of hypergraph trees and long subdivisions

The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and Rödl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and powers of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers tha…

图与超图中单调路径的Ramsey问题

Ramsey problems for monotone paths in graphs and hypergraphs

本文研究图与超图中单调路径的有序Ramsey数，证明了Mubayi和Suk的两个猜想，改进了Balko等人的界，并给出Erdős-Rado典型Ramsey定理的颜色单调版本。

Ramsey numbers of Berge-hypergraphs and related structures

For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a Berge-$G$, denoted by $BG$, if there exists a bijection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. Let the Ramsey number $R^r(BG,BG)$ be the smallest integer $n$ such that for any $2$-edge-coloring of a complete $r$-uniform hypergraph on $n$ vertices, there is a monochromatic Berge-$G$ subhypergraph. In this paper, we show that the 2-color Ramsey number of Berge cliques is linear. In particular, we show that $R^3(BK_s, BK_t) = s+t-3$ for $s,t \geq 4$ and $\max(s,t) \geq 5$ where $BK_n$ is a Be…

The Ramsey number of loose paths in 3-uniform hypergraphs

Recently, asymptotic values of 2-color Ramsey numbers for loose cycles and also loose paths were determined. Here we determine the 2-color Ramsey number of 3-uniform loose paths when one of the paths is significantly larger than the other: for every $n\geq \Big\lfloor\frac{5m}{4}\Big\rfloor$, we show that $$R(\mathcal{P}^3_n,\mathcal{P}^3_m)=2n+\Big\lfloor\frac{m+1}{2}\Big\rfloor.$$

围长拉姆齐定理

The girth Ramsey theorem

该研究证明了任意超图在给定颜色数下存在同围长的拉姆齐超图，且对线性超图可构造局部为副本森林的拉姆齐超图。

稀疏超图的Ramsey数和Turán数

Ramsey and Turán numbers of sparse hypergraphs

本文定义超图的骨架退化性，证明其控制超图Turán数和Ramsey数，并给出线性Ramsey数及Turán数上下界，使用依赖随机选择技术。

Hypergraph Ramsey numbers of cliques versus stars

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…

A note on the Erdős-Hajnal hypergraph Ramsey problem

We show that there is an absolute constant $c>0$ such that the following holds. For every $n > 1$, there is a 5-uniform hypergraph on at least $2^{2^{cn^{1/4}}}$ vertices with independence number at most $n$, where every set of 6 vertices induces at most 3 edges. The double exponential growth rate for the number of vertices is sharp. By applying a stepping-up lemma established by the first two authors, analogous sharp results are proved for $k$-uniform hypergraphs. This answers the penultimate open case of a conjecture in Ramsey theory posed by Erdős and Hajnal in 1972.

Degrees in oriented hypergraphs and sparse Ramsey theory

Let $G$ be an $r$-uniform hypergraph. When is it possible to orient the edges of $G$ in such a way that every $p$-set of vertices has some $p$-degree equal to $0$? (The $p$-degrees generalise for sets of vertices what in-degree and out-degree are for single vertices in directed graphs.) Caro and Hansberg asked if the obvious Hall-type necessary condition is also sufficient. Our main aim is to show that this is true for $r$ large (for given $p$), but false in general. Our counterexample is based on a new technique in sparse Ramsey theory that may be of independent interest.

Degrees in oriented hypergraphs and Ramsey p-chromatic number

The family $D(k,m)$ of graphs having an orientation such that for every vertex $v \in V(G)$ either (outdegree) $°^+(v) \le k$ or (indegree) $°^-(v) \le m$ have been investigated recently in several papers because of the role $D(k,m)$ plays in the efforts to estimate the maximum directed cut in digraphs and the minimum cover of digraphs by directed cuts. Results concerning the chromatic number of graphs in the family $D(k,m)$ have been obtained via the notion of $d$-degeneracy of graphs. In this paper we consider a far reaching generalization of the family $D(k,m)$, in a complementary form, int…

Anti-Ramsey number of matchings in $r$-partite $r$-uniform hypergraphs

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…

Ramsey numbers of sparse hypergraphs

We give a short proof that any k-uniform hypergraph H on n vertices with bounded degree Δhas Ramsey number at most c(Δ, k)n, for an appropriate constant c(Δ, k). This result was recently proved by several authors, but those proofs are all based on applications of the hypergraph regularity method. Here we give a much simpler, self-contained proof which uses new techniques developed recently by the authors together with an argument of Kostochka and Rödl. Moreover, our method demonstrates that, for k \geq 4, c(Δ, k) \leq 2^{2^{\Ddots^{2^{c Δ}}}}, where the tower is of height k and the constant c …

Ramsey theory constructions from hypergraph matchings

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…

Ramsey numbers for multiple copies of hypergraphs

In this paper, for sufficiently large $n$ we determine the Ramsey number $R(G,nH)$ where $G$ is a $k$-uniform hypergraph with the maximum independent set that intersects each of the edges in $k-1$ vertices and $H$ is a $k$-uniform hypergraph with a vertex so that the hypergraph induced by the edges containing this vertex is a star. There are several examples for such $G$ and $H$, among them are any disjoint union of $k$-uniform hypergraphs involving loose paths, loose cycles, tight paths, tight cycles with a multiple of $k$ edges, stars, Kneser hypergraphs and complete $k$-uniform $k$-partite …

圈、路和超图的广义Ramsey数

Generalized Ramsey numbers of cycles, paths, and hypergraphs

本文研究了超图的广义Ramsey数，在完全图、完全二部图及完全超图等设定下，针对圈、路、团和紧圈等子图结构，得出了若干渐近紧确的颜色数界限。

On ordered Ramsey numbers of tripartite 3-uniform hypergraphs

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…

非对角超图拉姆齐数何时是多项式级的？

When are off-diagonal hypergraph Ramsey numbers polynomial?

该研究针对拉姆齐理论中3-图非对角超图拉姆齐数何时随n多项式增长的开放问题，证明了紧连通或至多两个紧分支的3-图满足该性质当且仅当它包含于边的迭代膨胀中，推进了相关问题研究。

Diagonal Ramsey numbers of loose cycles in uniform hypergraphs

A $k$-uniform loose cycle $\mathcal{C}_n^k$ is a hypergraph with vertex set $\{v_1,v_2,\ldots,v_{n(k-1)}\}$ and with the set of $n$ edges $e_i=\{v_{(i-1)(k-1)+1},v_{(i-1)(k-1)+2},\ldots,v_{(i-1)(k-1)+k}\}$, $1\leq i\leq n$, where we use mod $n(k-1)$ arithmetic. The Ramsey number $R(\mathcal{C}^k_n,\mathcal{C}^k_n)$ is asymptotically $\frac{1}{2}(2k-1)n$ as has been proved by Gyárfás, Sárközy and Szemerédi. In this paper, we investigate to determining the exact value of diagonal Ramsey number of $\mathcal{C}^k_n$ and we show that for $n\geq 2$ and $k\geq 8$ $$R(\mathcal{C}^k_n,\mathcal{C}^k_n)=…

3-均匀超图中星形超图的反拉姆齐数

Anti-Ramsey Number of Stars in 3-uniform hypergraphs

本文研究了3-均匀超图中k-星形超图的反拉姆齐数，确定了特定条件下该数的精确值。

关于紧树-完全超图的拉姆齐数

On tight tree-complete hypergraph Ramsey numbers

本文扩展了Chvátal的树-完全图拉姆齐数结论，证明了对于3和4-一致非平凡紧树，其超图拉姆齐数满足 $R(T,n)=Θ(n^{r-1})$，主要贡献在于提出了受Cooper-Mubayi构造启发的4-一致构造。

超图Ramsey数的准多项式增长率

Hypergraph Ramsey numbers with quasipolynomial growth rate

研究提出首个准多项式增长的超图Ramsey数实例，揭示其既非多项式也非指数增长的特性。

On Ramsey numbers of 3-uniform Berge cycles

For an arbitrary graph $G$, a hypergraph $\mathcal{H}$ is called Berge-$G$ if there is a bijection $Φ:E(G)\longrightarrow E( \mathcal{H})$ such that for each $e\in E(G)$, we have $e\subseteq Φ(e)$. We denote by $\mathcal{B}^rG$, the family of $r$-uniform Berge-$G$ hypergraphs. For families $\mathcal{H}_1, \mathcal{H}_2,\ldots, \mathcal{H}_t$ of $r$-uniform hypergraphs, the Ramsey number $R(\mathcal{H}_1, \mathcal{H}_2,\ldots, \mathcal{H}_t)$ is the smallest integer $n$ such that in every $t$-hyperedge coloring of $\mathcal{K}_{n}^r$ there is a monochromatic copy of a hypergraph in $\mathcal{H}…

有界度超图的Ramsey下界

Two-color Ramsey lower bounds for bounded degree hypergraphs

研究提出一种k-超图，其最大度不超过Δ，且Ramsey数r(H)达到塔函数级别，推动了Conlon等人2009年问题的进展。

Linear Ramsey numbers for bounded-degree hypergraphs

We show that the Ramsey number is linear for every uniform hypergraph with bounded-degree. This is a hypergraph extension of the famous theorem for ordinary graphs which Chvátal et al. showed in 1983. Our proof is simple, contains the multicolor case, and provides a strong embedding lemma. It shows the potential of a new hypergraph regularity lemma by the author.

一致超图中松散路径与松散圈的anti-Ramsey数

Anti-Ramsey numbers of loose paths and cycles in uniform hypergraphs

本文研究一致超图中松散路径与松散圈的anti-Ramsey数，对所有k≥4和r≥3给出了其精确值。

New lower bounds for hypergraph Ramsey numbers

The Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple among them is blue. We prove the following new lower bounds for 4-uniform hypergraph Ramsey numbers: $$r_4(5,n) > 2^{n^{c\log n}} \qquad \hbox{ and } \qquad r_4(6,n) > 2^{2^{cn^{1/5}}},$$ where $c$ is an absolute positive constant. This substantially improves the previous best bounds of $2^{n^{c\log\log n}}$ and $2^{n^{c\log n}}$, respectively. Using previously …

On the Ramsey-Turán numbers of graphs and hypergraphs

Let t be an integer, f(n) a function, and H a graph. Define the t-Ramsey-Turán number of H, RT_t(n, H, f(n)), to be the maximum number of edges in an n-vertex, H-free graph G where f(n) is larger than the maximum number of vertices in a $K_t$-free induced subgraph of G. Erdős, Hajnal, Simonovits, Sós, and Szemerédi posed several open questions about RT_t(n,K_s,o(n)), among them finding the minimum s such that $RT_t(n,K_{t+s},o(n)) = Ω(n^2)$, where it is easy to see that $RT_t(n,K_{t+1},o(n)) = o(n^2)$. In this paper, we answer this question by proving that $RT_t(n,K_{t+2},o(n)) = Ω(n^2)$; our …

The Ramsey Number of Fano Plane Versus Tight Path

The hypergraph Ramsey number of two $3$-uniform hypergraphs $G$ and $H$, denoted by $R(G,H)$, is the least integer $N$ such that every red-blue edge-coloring of the complete $3$-uniform hypergraph on $N$ vertices contains a red copy of $G$ or a blue copy of $H$. The Fano plane $\mathbb{F}$ is the unique 3-uniform hypergraph with seven edges on seven vertices in which every pair of vertices is contained in a unique edge. There is a simple construction showing that $R(H,\mathbb{F}) \ge 2(v(H)-1) + 1.$ Hypergraphs $H$ for which the equality holds are called $\mathbb{F}$-good. Conlon asked to dete…

线性超图的非对角 Ramsey 数

Off-Diagonal Ramsey Numbers for Linear Hypergraphs

本文研究线性超图的非对角 Ramsey 数，证明在均匀度 $k\ge 4$ 时，存在线性 $k$-均匀超图使 Ramsey 数具有远高于多项式的增长率。

A Ramsey-type result for geometric l-hypergraphs

Let n \geq l \geq 2 and q \geq 2. We consider the minimum N such that whenever we have N points in the plane in general position and the l-subsets of these points are colored with q colors, there is a subset S of n points all of whose l-subsets have the same color and furthermore S is in convex position. This combines two classical areas of intense study over the last 75 years: the Ramsey problem for hypergraphs and the Erd\H os-Szekeres theorem on convex configurations in the plane. For the special case l = 2, we establish a single exponential bound on the minimum N, such that every complete …

多部图与超图的奇 Ramsey 数

Odd Ramsey numbers of multipartite graphs and hypergraphs

本文研究多部图与超图的奇 Ramsey 数，推广已有结果并确定两类结构的渐近值，其中超图情形属首次。

On generalized Ramsey numbers for 3-uniform hypergraphs

The well-known Ramsey number $r(t,u)$ is the smallest integer $n$ such that every $K_t$-free graph of order $n$ contains an independent set of size $u$. In other words, it contains a subset of $u$ vertices with no $K_2$. Erd{\H o}s and Rogers introduced a more general problem replacing $K_2$ by $K_s$ for $2\le s<t$. Extending the problem of determining Ramsey numbers they defined the numbers $$ f_{s,t}(n)=\min \big{\{} \max \{|W| : W\subseteq V(G) \text{and} G[W] \text{contains no} K_s\}\big{\}}, $$ where the minimum is taken over all $K_t$-free graphs $G$ of order $n$. In this note, we study …

A note on induced Ramsey numbers

The induced Ramsey number $r_{\mathrm{ind}}(F)$ of a $k$-uniform hypergraph $F$ is the smallest natural number $n$ for which there exists a $k$-uniform hypergraph $G$ on $n$ vertices such that every two-coloring of the edges of $G$ contains an induced monochromatic copy of $F$. We study this function, showing that $r_{\mathrm{ind}}(F)$ is bounded above by a reasonable power of $r(F)$. In particular, our result implies that $r_{\mathrm{ind}}(F) \leq 2^{2^{ct}}$ for any $3$-uniform hypergraph $F$ with $t$ vertices, mirroring the best known bound for the usual Ramsey number. The proof relies on a…

Ramsey-type numbers involving graphs and hypergraphs with large girth

A question of Erdős asks if for every pair of positive integers $r$ and $k$, there exists a graph $H$ having $\textrm{girth}(H)=k$ and the property that every $r$-colouring of the edges of $H$ yields a monochromatic cycle $C_k$. The existence of such graphs was confirmed by the third author and Ruciński. We consider the related numerical problem of determining the smallest such graph with this property. We show that for integers $r$ and $k$, there exists a graph $H$ on $R^{10k^2} k^{15k^3}$ vertices (where $R = R(C_k;r)$ is the $r$-colour Ramsey number for the cycle $C_k$) having $\textrm{girt…

Independent sets in hypergraphs with a forbidden link

We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding open problem of Erdős and Hajnal in Ramsey theory. We further extend this result to prove tight bounds on various other hypergraph Ramsey numbers.

Algebraic Properties of a Hypergraph Lifting Map

Recent work in hypergraph Ramsey theory has involved the introduction of a "lifting map" that associates a certain $3$-uniform hypergraph to a given graph, bounding cliques in a predictable way. In this paper, we interpret the lifting map as a linear transformation. This interpretation allows us to use algebraic techniques to prove several structural properties of the lifting map, culminating in new lower bounds for certain $3$-uniform hypergraph Ramsey numbers.

缓慢增长超图的对角外拉姆齐数

Off-diagonal Ramsey numbers for slowly growing hypergraphs

本文研究缓慢增长超图的对角外拉姆齐数，证明对任意非k-部缓慢增长k-一致超图F，r(F,n)至少为n^k除以对数因子的量级，并首次确定三元系F_5的对角外拉姆齐数多项式幂次，构造综合运用伪随机图、鞅与超图容器方法。

Independent sets in hypergraphs and Ramsey properties of graphs and the integers

Many important problems in combinatorics and other related areas can be phrased in the language of independent sets in hypergraphs. Recently Balogh, Morris and Samotij, and independently Saxton and Thomason developed very general container theorems for independent sets in hypergraphs; both of which have seen numerous applications to a wide range of problems. In this paper we use the container method to give relatively short and elementary proofs of a number of results concerning Ramsey (and Turán) properties of (hyper)graphs and the integers. In particular: (i) We generalise the random Ramsey …

Ramsey properties of algebraic graphs and hypergraphs

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…

Hypergraph Ramsey numbers

The Ramsey number r_k(s,n) is the minimum N such that every red-blue coloring of the k-tuples of an N-element set contains either a red set of size s or a blue set of size n, where a set is called red (blue) if all k-tuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for r_k(s,n) for k \geq 3 and s fixed. In particular, we show that r_3(s,n) \leq 2^{n^{s-2}\log n}, which improves by a factor of n^{s-2}/ polylog n the exponent of the previous upper bound of Erdos and Rado from 1952. We also obtain a …

Ramsey problems for Berge hypergraphs

For a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $G$ (or a Berge-$G$ in short), if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for each $e \in E(G)$ we have $e \subseteq f(e)$. We denote the family of $r$-uniform hypergraphs that are Berge copies of $G$ by $B^rG$. For families of $r$-uniform hypergraphs $\mathbf{H}$ and $\mathbf{H}'$, we denote by $R(\mathbf{H},\mathbf{H}')$ the smallest number $n$ such that in any blue-red coloring of $\mathcal{K}_n^r$ (the complete $r$-uniform hypergraph on $n$ vertices) there is a monochromatic blue copy of a hypergrap…

强Ramsey游戏中3一致超图的绘制策略

Drawing strategies in Strong Ramsey games for 3-uniform hypergraphs

本文研究强Ramsey游戏，给出无限族3一致超图使得后手玩家有平局策略。

The size-Ramsey number of 3-uniform tight paths

Given a hypergraph $H$, the size-Ramsey number $\hat{r}_2(H)$ is the smallest integer $m$ such that there exists a graph $G$ with $m$ edges with the property that in any colouring of the edges of $G$ with two colours there is a monochromatic copy of $H$. We prove that the size-Ramsey number of the $3$-uniform tight path on $n$ vertices $P^{(3)}_n$ is linear in $n$, i.e., $\hat{r}_2(P^{(3)}_n) = O(n)$. This answers a question by Dudek, Fleur, Mubayi, and Rödl for $3$-uniform hypergraphs [On the size-Ramsey number of hypergraphs, J. Graph Theory 86 (2016), 417-434], who proved $\hat{r}_2(P^{(3)}…

非对角超图拉姆齐数的新边界

New bounds for off-diagonal hypergraph Ramsey numbers

该研究针对Conlon等人的拉姆齐数猜想，证明了至多5个紧分支的3-图满足该猜想，推导了其与线性3-图特例的等价性，给出可向前双化3-图的拟多项式上界、非向前层状3-图的指数下界，并分类了链族的拉姆齐增长性。

超图反拉姆齐定理

Hypergraph anti-Ramsey theorems

本文研究了超图反拉姆齐问题，针对较小均匀性超图的膨胀建立了消去型结果，并据此改进了Erdős--Simonovits--Sós的一般界，同时确定了特定图类膨胀在大n时的精确值，将完全图结果扩展至超图领域。

关于多色Ramsey数与超图的子集着色

On Multicolour Ramsey Numbers and Subset-Colouring of Hypergraphs

本文改进了多色超图Ramsey数的已知下界，并研究了使超图中大超边非单色的子集着色问题，基于色数给出了所需颜色数的上下界。

随机扰动超图的 Ramsey 性质

Ramsey properties of randomly perturbed hypergraphs

本文研究随机扰动 3-一致超图的 Ramsey 性质，利用融合强弱形式的新超图正则性引理及元组引理，确定了诱导扩展团两色 Ramsey 性质所需随机超边的阈值。

Hypergraph Ramsey numbers: tight cycles versus cliques

For $s \ge 4$, the 3-uniform tight cycle $C^3_s$ has vertex set corresponding to $s$ distinct points on a circle and edge set given by the $s$ cyclic intervals of three consecutive points. For fixed $s \ge 4$ and $s \not\equiv 0$ (mod 3) we prove that there are positive constants $a$ and $b$ with $$2^{at}<r(C^3_s, K^3_t)<2^{bt^2\log t}.$$ The lower bound is obtained via a probabilistic construction. The upper bound for $s>5$ is proved by using supersaturation and the known upper bound for $r(K_4^{3}, K_t^3)$, while for $s=5$ it follows from a new upper bound for $r(K_5^{3-}, K_t^3)$ that we de…

On the cover Ramsey number of Berge hypergraphs

For a fixed set of positive integers $R$, we say $\mathcal{H}$ is an $R$-uniform hypergraph, or $R$-graph, if the cardinality of each edge belongs to $R$. An $R$-graph $\mathcal{H}$ is \emph{covering} if every vertex pair of $\mathcal{H}$ is contained in some hyperedge. For a graph $G=(V,E)$, a hypergraph $\mathcal{H}$ is called a \textit{Berge}-$G$, denoted by $BG$, if there exists an injection $f: E(G) \to E(\mathcal{H})$ such that for every $e \in E(G)$, $e \subseteq f(e)$. In this note, we define a new type of Ramsey number, namely the \emph{cover Ramsey number}, denoted as $\hat{R}^R(BG_1…