AI 中文总结
本文针对最大临界t-相交超图的Frankl猜想，将成立的k阈值从d⁴改进为30d²，借助Frankl的固定边分解与Füredi的伪向日葵方法完成证明。
AI 中文摘要
设k>t≥1为整数，令d=k−t。k-均匀超图F称为t-相交的，若任意两条边的交集至少含t个顶点；称为t-临界的，若其最小t-横贯大小为k。Frankl证明当k≥d⁴时，|F|≤C(k+d,d)，等号仅对k+d个顶点上的完全k-图成立，并猜想当k>cd²（c为某常数）时结论仍成立。本文对c=30证实该猜想，证明依赖Frankl的固定边分解与Füredi的伪向日葵方法。
英文摘要
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.
