Concise proofs concerning the size and structure of large intersecting \(k\)-graphs

Abstract

Let \(n > 2k \geq 4\) and let \(\mathcal F \subset {[n]\choose k}\) be an intersecting \(k\)-graph on \(n\) vertices, that is, \(F\cap F' \neq \emptyset\) for all \(F, F' \in \mathcal F\). By the Erd\H{o}s--Ko--Rado Theorem \(|\mathcal F| \leq { n - 1\choose k - 1}\) with equality holding only for the full-star, \(\mathcal S_x\), the family of all \(k\)-sets containing the vertex \(x\). If we exclude stars, that is, subfamilies of \(\mathcal S_x\) then \(|\mathcal F| \leq { n - 1\choose k - 1} - { n - k - 1\choose k - 1} + 1\) was proved by Hilton and Milner who determined the families attaining equality as well. Half a century later Han and Kohayakawa determined the next largest families. Then very recently Huang and Peng determined the fourth largest families. That is, they determined the largest families assuming that \(\mathcal F\) is not a star and it is not contained neither in the Hilton--Milner families nor in the Han--Kohayakawa families. In the present paper we provide a unified simple proof for these two theorems as well as solve the corresponding problem for \(t \)-intersecting families \((|F \cap F'| \geq t)\) albeit only for \(n > t + \max \{4t(k - t + 1)^2, 2(t + 1)^2\}\).