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 $k\ge d^4$,$|\mathcal F|\le \binom{k+d}{d},$ 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.