An intersection problem with 6 extremes (Q793026)

The author proves the following theorem: Let \(X\) be a finite set of cardinality \(n\) and let \(\mathcal F\) be a family of \(r\)-subsets of \(X\). Suppose that any two members of \(\mathcal F\) intersect and for some \(3/5<c<2/3,\) for \(n>n_0(r,c),\) and for every \(x\in X\), \(| \{F;x\in F\in {\mathcal F}\}| \leq c| {\mathcal F}|\) holds. Then \[ | {\mathcal F}| \leq 10\binom{n-5}{r-3} + 5 \binom{n-5}{r-4} + \binom{n-5}{r-5}. \] Extremal families are also described. This result is an improvement of a theorem of \textit{P. Frankl} [J. Comb. Theory, Ser. A 24, 146--161 (1978; Zbl 0384.05002)].