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

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 $k\ge d^4$,$\mathrm{\mid }\mathcal{F}\mathrm{\mid }\le \left(\genfrac{}{}{0px}{}{k+d}{d}\right),$ 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.