A hierarchy of maximal intersecting triple systems

Abstract

We reach beyond the celebrated theorems of Erdȍs-Ko-Rado and Hilton-Milner, and a recent theorem of Han-Kohayakawa, and determine all maximal intersecting triples systems. It turns out that for each $n\geq 7$ there are exactly 15 pairwise non-isomorphic such systems (and 13 for n=6). We present our result in terms of a hierarchy of Turán numbers $\operatorname{ex}^{(s)}(n; M_2^{3})$, $s\geq 1$, where $M_2^{3}$ is a pair of disjoint triples. Moreover, owing to our unified approach, we provide short proofs of the above mentioned results (for triple systems only). The triangle $C_3$ is defined as $C_3=\{\{x_1,y_3,x_2\},\{x_1,y_2,x_3\},\{x_2,y_1,x_3\}\}$. Along the way we show that the largest intersecting triple system $H$ on $n\geq 6$ vertices, which is not a star and is triangle-free, consists of $\max\{10,n\}$ triples. This facilitates our main proof's philosophy which is to assume that $H$ contains a copy of the triangle and analyze how the remaining edges of $H$ intersect that copy.