An improved range for the maximum critically $t$-intersecting hypergraphs

Lu Lu; Rongrong Lu; Qifan Wang; Tingzeng Wu

combinatorics

An improved range for the maximum critically $t$ -intersecting hypergraphs

arXiv:2607.28253

summary

The paper proves that for k-uniform hypergraphs that are t‑intersecting and t‑critical, the maximum number of edges is bounded by $\binom{k+d}{d}$ when k > 30·d², confirming Frankl’s conjecture for this constant.

Abstract

Let $k > t \geq 1$ be integers and set $d = k - t$ . A $k$ -uniform hypergraph $F$ 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 $k \geq d^{4}$ , $∣ F ∣ \leq (d k + d),$ 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 FÃ¼redi's pseudo-sunflower method.

Topics & keywords

#extremal hypergraph theory #intersecting families #t‑intersecting #critical hypergraphs #combinatorial boundsk‑uniform hypergrapht‑intersectingt‑criticalt‑transversalFrankl's conjecturepseudo‑sunflower method

An improved range for the maximum critically t-intersecting hypergraphs

Abstract

Topics & keywords

An improved range for the maximum critically $t$ -intersecting hypergraphs