Guided by the connections between hypergraphs and exterior algebras, we study Turán and Ramsey type problems for alternating multilinear maps. This study lies at the intersection of combinatorics, group theory, and algebraic geometry, and has origins in the works of Lovász (Proc. Sixth British Combinatorial Conf., 1977), Buhler, Gupta, and Harris (J. Algebra, 1987), and Feldman and Propp (Adv. Math., 1992).
Our main result is a Ramsey theorem for alternating bilinear maps. Given $s, t\in \mathbb{N}$, $s, t\geq 2$, and an alternating bilinear map $f:V\times V\to U$ with $\dim(V)=s\cdot t^4$, we show that there exists either a dimension-$s$ subspace $W\leq V$ such that $\dim(f(W, W))=0$, or a dimension-$t$ subspace $W\leq V$ such that $\dim(f(W, W))=\binom{t}{2}$. This result has natural group-theoretic (for finite $p$-groups) and geometric (for Grassmannians) implications, and leads to new Ramsey-type questions for varieties of groups and Grassmannians.
We determine Turán numbers for the family of 3-uniform minimal paths of length four \emph{for all $n$}. We also establish the second and third order Turán numbers and use them to compute the corresponding Ramsey numbers for up to four colors.
We study Turán and Ramsey-type problems on edge-colored graphs. An edge-colored graph is called {\em $\varepsilon$-balanced} if each color class contains at least an $\varepsilon$-proportion of its edges. Given a family $\mathcal{F}$ of edge-colored graphs, the Ramsey function $R(\varepsilon, \mathcal{F})$ is the smallest $n$ for which any $\varepsilon$-balanced $K_n$ must contain a copy of an $F\in\mathcal{F}$, and the Turán function $\mathrm{ex}(\varepsilon, n, \mathcal{F})$ is the maximum number of edges in an $n$-vertex $\varepsilon$-balanced graph which avoids all of $\mathcal{F}$. In thi…
Academy of Mathematics and Systems Science, Chinese Academy of Sciences（中国科学院数学与系统科学研究院）；School of Mathematical Sciences, Zhejiang University（浙江大学数学学院）
We say that a subgraph $F$ of a graph $G$ is singular if the degrees $d_G(v)$ are all equal or all distinct for the vertices $v\in V(F)$. The singular Ramsey number Rs$(F)$ is the smallest positive integer $n$ such that, for every $m\geq n$, in every edge 2-coloring of $K_m$, at least one of the color classes contains $F$ as a singular subgraph. In a similar flavor, the singular Turán number Ts$(n,F)$ is defined as the maximum number of edges in a graph of order $n$, which does not contain $F$ as a singular subgraph. In this paper we initiate the study of these extremal problems. We develop me…
In this paper we study Turán and Ramsey numbers in linear triple systems, defined as $3$-uniform hypergraphs in which any two triples intersect in at most one vertex. A famous result of Ruzsa and Szemerédi is that for any fixed $c>0$ and large enough $n$ the following Turán-type theorem holds. If a linear triple system on $n$ vertices has at least $cn^2$ edges then it contains a {\em triangle}: three pairwise intersecting triples without a common vertex. In this paper we extend this result from triangles to other triple systems, called {\em $s$-configurations}. The main tool is a generalizatio…
设r是一个整数，f（n）是一个函数，H是一个图.由Erdés，Hajnal，Sós和Szemerédi引入的H的r-Ramsey-Turán数RT_r（n，H，f（n））定义为n-顶点无H图G的最大边数，其中α_r（G）<= f（n），其中α_r（G）表示G的K_r-独立数.本文利用高维单位球的等周性质，构造了对任意2 <= s <= r，RT_r（n，K_{r+s}，o（n））有下界的图.这些构造对于无限族的r和s对是尖锐的。唯一以前的尖锐建设是由Bollobás和埃尔德\Hos为r = s = 2。
VeraT.索斯要求确定一个给定的整数$t\ge 3$和一个给定的正实数$δ$的渐近极大边密度$f_t(δ)$，一个$n$ -顶点图可以提供它既不包含一个完整的图形$K_t$，也不是一个独立集大小$δn$。基于Fox，Loh和Zhao最近的工作[The critical window for the classical Ramsey-Turán problem，Combinatorica 35（2015），435-476]，我们证明了如果$δ$足够小（在某种意义上取决于$t$），则\[ f_t(δ)= \begin{cases} \frac{3t-10}{3t-4}+δ-δ^2 & \text{ if $t$ is even,} \cr \frac{t-3}{t-1}+δ& \text{ if $t$ is odd.} \end{cases} \]
Given a graph $H$ and a function $f:\mathbb{Z}^+ \longrightarrow \mathbb{Z}^+ $, the Ramsey-Turán number of $H$ and $f$, denoted by $RT(n, H, f(n))$, is the maximum number of edges a graph $G$ on $n$ vertices can have, which does not contain $H$ as a subgraph and also does not contain a set of $f(n)$ independent vertices. Let $r$ be a positive integer. In 1969, Erdős and Sós proved that $RT(n,K_{2r+1},o(n))=\frac{n^2}{2}(1-\frac{1}{r})+o(n^2)$. Let $F_k(2r+1)$ denote the graph consisting of $k$ copies of complete graphs $K_{2r+1}$ sharing exactly one vertex. In this paper, we show that $RT(n,F…
Given positive integers $n\ge s$, we let ${\mathrm{ex}}(n,s)$ denote the maximum number of edges in a triangle-free graph $G$ on $n$ vertices with $α(G)\le s$. In the early sixties Andrásfai conjectured that for $n/3<s<n/2$ the function ${\mathrm{ex}}(n, s)$ is piecewise quadratic with critical values at $s/n={k}/({3k-1})$. We confirm that this is indeed the case whenever $s/n$ is slightly larger than a critical value, thus determining ${\mathrm{ex}}(n,s)$ for all $n$ and $s$ such that $s/n\in [{k}/({3k-1}), {k}/({3k-1})+γ_k]$, where $γ_k=Θ(k^{-6})$.
The anti-Ramsey problem was introduced by Erdős, Simonovits and Sós in 1970s. The anti-Ramsey number of a hypergraph $\mathcal{H}$, $ar(n,s, \mathcal{H})$, is the smallest integer $c$ such that in any coloring of the edges of the $s$-uniform complete hypergraph on $n$ vertices with exactly $c$ colors, there is a copy of $\mathcal{H}$ whose edges have distinct colors. In this paper, we determine the anti-Ramsey numbers of linear paths and loose paths in hypergraphs for sufficiently large $n$, and give bounds for the anti-Ramsey numbers of Berge paths. Similar exact anti-Ramsey numbers are obtai…
One of the oldest results in modern graph theory, due to Mantel, asserts that every triangle-free graphs on $n$ vertices has at most $\lfloor n^2/4\rfloor$ edges. About half a century later Andrásfai studied dense triangle-free graphs and proved that the largest triangle-free graphs on $n$ vertices without independent sets of size $αn$, where $2/5\le α< 1/2$, are blow-ups of the pentagon. More than 50 further years have elapsed since Andrásfai's work. In this article we make the next step towards understanding the structure of dense triangle-free graphs without large independent sets. Notably,…
We call a $4$-cycle in $K_{n_{1}, n_{2}, n_{3}}$ multipartite, denoted by $C_{4}^{\text{multi}}$, if it contains at least one vertex in each part of $K_{n_{1}, n_{2}, n_{3}}$. The Turán number $\text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})$ $\bigg($ respectively, $\text{ex}(K_{n_{1},n_{2},n_{3}},\{C_{3}, C_{4}^{\text{multi}}\})$ $\bigg)$ is the maximum number of edges in a graph $G\subseteq K_{n_{1},n_{2},n_{3}}$ such that $G$ contains no $C_{4}^{\text{multi}}$ $\bigg($ respectively, $G$ contains neither $C_{3}$ nor $C_{4}^{\text{multi}}$ $\bigg)$. We call a $C^{multi}_4$ rainbow if a…
The $(m,n)$-online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge on each turn and Painter immediately paints it red or blue. Builder's goal is to force Painter to create either a red $K_m$ or a blue $K_n$ using as few turns as possible. The online Ramsey number $\tilde{r}(m,n)$ is the minimum number of edges Builder needs to guarantee a win in the $(m,n)$-online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvement \[ \tilde{r}(n,n) \…
We introduce a graph Ramsey game called Ramsey, Paper, Scissors. This game has two players, Proposer and Decider. Starting from an empty graph on $n$ vertices, on each turn Proposer proposes a potential edge and Decider simultaneously decides (without knowing Proposer's choice) whether to add it to the graph. Proposer cannot propose an edge which would create a triangle in the graph. The game ends when Proposer has no legal moves remaining, and Proposer wins if the final graph has independence number at least $s$. We prove a threshold phenomenon exists for this game by exhibiting randomized st…
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 …
HUN-REN Alfréd Rényi Institute of Mathematics（HUN-REN 阿尔弗雷德·雷尼数学研究所）；Academy of Plateau Science and Sustainability, and School of Mathematics and Statistics, Qinghai Normal University（青海师范大学高原科学与可持续发展研究院及数学与统计学院）
In 1964, Erdős, Hajnal and Moon introduced a saturation version of Turán's classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of this problem in other settings. We prove a saturation version of the Erdős-Szekeres theorem about monotone subsequences and saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets. We also consider semisaturation problems, wherein we allow the…
An important theme of recent research in Ramsey theory has been establishing pseudorandomness properties of Ramsey graphs. An $N$-vertex graph is called $C$-Ramsey if it has no homogeneous set of size $C\log N$. A theorem of Bukh and Sudakov, solving a conjecture of Erdős, Faudree and Sós, shows that any $C$-Ramsey $N$-vertex graph contains an induced subgraph with $Ω_C(N^{1/2})$ distinct degrees. We improve this to $Ω_C(N^{2/3})$, which is tight up to the constant factor. We also show that any $N$-vertex graph with $N > (k-1)(n-1)$ and $n\geq n_0(k) = Ω(k^9)$ either contains a homogeneous set…
