A universal bound concerning \(t\)-intersecting families (Q6061164)

A family of sets \({\mathcal{F}}\subset\binom{[n]}{k}\) is \textit{\(t\)-intersecting\/} if, for all \(F_1,F_2\in{\mathcal{F}}\), \(\left\vert F_1\cap F_2\right\vert \ge t\). Applying a lemma from \textit{P. Frankl} [Combinatorics, Keszthely 1976, Colloq. Math. Soc. János Bolyai 18, 365--375 (1978; Zbl 0401.05001)] the author provides a short inductive proof of the following Theorem: \textit{Let \(n\ge k\ge t \ge s\ge1\) and let \({\mathcal{F}}\subseteq\binom{[n]}{k}\) be \(t\)-intersecting. Suppose further that \(n\ge(s+1)k-st+1\). Then \(\vert {\mathcal{F}}\vert \le\binom{n-s}{k-t}\).\/} This reduces to the Erdős-Ko-Rado Theorem by \textit{P. Erdős} et al. [Q. J. Math., Oxf. II. Ser. 12, 313--320 (1961; Zbl 0100.01902)] when \(s=t\). (The case \(s=1\) was proved earlier by the author, with a different argument, \textit{P. Frankl} [Eur. J. Comb. 87, Article ID 103134, 3 p. (2020; Zbl 1439.05216)].