Stability of intersecting families (Q6081103)

The classical Erdős-Ko-Rado theorem determines the size of the largest intersecting family \(\mathcal F \subset \binom{[n]}{k}\), which turns out to be a trivial family/ full star (a family with an element contained in all sets), except for some sporadic cases when \(n \le 2k\). \textit{A. J. W. Hilton} and \textit{E. C. Milner} [Q. J. Math., Oxf. II. Ser. 18, 369--384 (1967; Zbl 0168.26205)] determined the maximum size of an intersecting family which is not trivial (the intersection of all sets is the empty set), i.e., in essence, the second largest maximum intersecting family. In that line, \textit{J. Han} and \textit{Y. Kohayakawa} [Proc. Am. Math. Soc. 145, No. 1, 73--87 (2017; Zbl 1350.05169)] determined the third largest maximum intersecting family (MIF), and \textit{A. Kostochka} and \textit{D. Mubayi} [ibid. 145, No. 6, 2311--2321 (2017; Zbl 1358.05039)] the fourth largest MIF for \(n\) sufficiently large (in terms of \(k\)). In this paper, the authors extend the previous work, by giving a proof for the fourth largest MIF for every \(n \ge 2k+1.\) Hereby there is a difference for the range \(2k+1 \le n \le 3k-3\) and \(n \ge 3k-2.\) Let \(k\ge 4\) and \(\mathcal{H} \subseteq \binom{[n]}{k}\) be an intersecting family which is neither a subfamily of the extremal families from Erdős-Ko-Rado, Hilton and Milner [loc. cit.] or Han and Kohayakawa [loc. cit.]. Then for \(2k+1\le n\le 3k-3\), \[|\mathcal{H}| \le \binom{n-1}{k-1} -2\binom{n-k-1}{k-1} +\binom{n-k-3}{k-1}+2,\] For \(n\ge 3k-2\), \[|\mathcal{H}|\le \binom{n-1}{k-1} -\binom{n-k-1}{k-1} -\binom{n-k-2}{k-2} -\binom{n-k-3}{k-3}+3.\] The proof contains multiple lemmas and claims, as the authors also characterize the extremal families (for which there are two main constructions based on \(n \ge 3k-2\) or \(n \le 3k-3\), as well as some exceptional cases as well for \(k \in \{4,5\}\)).