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

Abstract

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 Füredi's pseudo-sunflower method.

Disclosure

“Lu Lu is supported by National Natural Science Foundation of China (No. 12371362). T. Wu was supported by the NSFC (No. 12261071) and NSF of Qinghai Province (No. 2025-ZJ-902T). Declaration of AI usage The authors acknowledge the use of AI tools during the exploratory stage of this project. All mathematical arguments and proofs in the final manuscript were checked and written by the authors. References [1] P. Erdős, C. Ko, and R. Rado, Intersection theorems for systems of fini”

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2026-08-23T19:14:01.787290Z

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