Let $k>t\ge 1$ be integers and set $d=k-t$. A $k$-uniform hypergraph $\mathcal F$ is called $t$-intersecting if any two edges intersect in at least $t$ vertices, and is called $t$-critical if its minimum $t$-transversal has size $k$. Frankl proved that, for $k\ge d^4$,$|\mathcal F|\le \binom{k+d}{d},$ with equality only for the complete $k$-graph on $k+d$ vertices, and conjectured that the same conclusion should hold when $k>c d^2$ for some constant $c$. In this paper we confirm this conjecture for $c=30$. The proof relies on Frankl's fixed-edge decomposition and F\"{u}redi's pseudo-sunflower method.
For a family $\mathcal{F}$ of $k$-graphs, $\ex_k(n,\mathcal{F})$ denotes the maximum number of edges in an $n$-vertex $\mathcal{F}$-free $k$-graph. Let $M_{s+1}^k$ denote a matching of size $s+1$ in $k$-uniform hypergraphs. Recently, Alon and Frankl (JCTB, 2024) determined $\ex_2(n,\{M_{s+1}^2,K_{\ell+1}\})$ for all $n...
For positive integers $d$ and $k$, let $n_k(d)$ be the maximum order of a graph of maximum degree at most $d$ and diameter at most $k$. We prove that $$ \lim_{d\to\infty}\frac{n_k(d)}{d^k}=1$$ for every fixed $k$, thereby resolving the asymptotic degree-diameter problem for fixed diameter and proving a conjecture of Bo...
Wouter Cames van Batenburg, Samuel Korsky· 0 citations
For an $n$-vertex graph $G$ and a permutation $\sigma$ of its vertex set, let $\sigma(G)$ denote the corresponding relabelling of $G$, and put $I_G(\sigma)=|E(G)\cap E(\sigma(G))|$. Let $f(n,k)$ be the minimum number of edges in an $n$-vertex graph for which $I_G(\sigma)\geq k$ for every $\sigma$. In his 1977 formulati...
For a tournament $T$, let $\omega(T)$ be the minimum clique number among the backedge graphs of $T$, and let $\chi(T)$ be its dichromatic number. We give a template-lifting construction. It turns a $k$-template into a regular, vertex-transitive Cayley tournament that is simultaneously $(k+1)$-$\omega$-critical and $(k+...
A subgraph $H$ of a $k$-connected graph $G$ is called \emph{$k$-removable} if $G-E(H)$ remains $k$-connected. Halin proved that every $k$-connected graph $G$ with $\delta(G)\ge k+1$ has a $k$-removable edge. We extend this result from a single edge to matchings of any prescribed size by showing that, for positive integ...
Put $q=r-1$, $t=k-1$, and $D=tq+1$. For an edge $e$ of a linear $C^r_{1,k}$-free $r$-uniform hypergraph, define \[ \delta_H(e)=\sum_{v\in e}\frac{1}{d_H(v)}-\frac{r}{D}. \] The defect satisfies $\delta_H(e)\ge 0$. At equality, every vertex of $e$ has degree $D$, and the petal trace at $e$ is a disjoint union of $t$ aff...
Mahesh Ramani· 0 citations
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