For positive integers $n\ge s>r$, let $T(n,s,r)$ denote the minimum number of edges in an $r$-uniform hypergraph on $n$ vertices such that every $s$-set of vertices contains at least one edge. A simple averaging argument shows that the ratio $T(n,s,r)/\binom nr$ is non-decreasing in $n$ and we denote its limit as $n\to...
Jun Gao, Pei-Ru Kuang, Oleg Pikhurko et al.· 0 citations
The local--global principle, which concerns the relationship between local structure and global parameters, has attracted considerable attention in extremal combinatorics over the past few decades. In this paper, we study how global density forces small dense subhypergraphs in uniform hypergraphs. For fixed $r\ge3$ and...
For an integer $n\ge0$, let $f(n)$ be the minimum number of subcubes of $\mathbb{Z}_3^n$ of the form $A_1\times\cdots\times A_n$, where $|A_i|=2$ for every $i$, whose union covers $\mathbb{Z}_3^n$. A simple counting argument gives $f(n)\ge(3/2)^n$, while $f(n)=O(n(3/2)^n)$ by random construction. We prove that $f(n)\le...
For graphs $F$ and $H$, the Ramsey number $R(F,H)$ is the minimum integer $N$ such that every $N$-vertex graph contains $F$ or its complement contains $H$. If $F$ is connected and $|F|\ge\sigma(H)$, a construction of Burr gives $R(F,H)\ge(\chi(H)-1)(|F|-1)+\sigma(H)$, where $\sigma(H)$ denotes the minimum order of a co...
Peiru Kuang, Yan Wang· 0 citations
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