A hierarchy of maximal intersecting triple systems

TL;DR

This paper classifies all maximal intersecting triple systems for sufficiently large n, extending classical theorems, and introduces a hierarchy of Turán numbers to unify and simplify proofs of these combinatorial structures.

Contribution

It provides a complete classification of maximal intersecting triples systems for n ≥ 6, introduces a hierarchy of Turán numbers, and offers unified, concise proofs of key results in the field.

Findings

Exactly 15 non-isomorphic maximal intersecting triples systems for n≥7

Largest non-star, triangle-free intersecting triple system has max{10, n} triples

Unified approach simplifies proofs of classical theorems in triple systems

Abstract

We reach beyond the celebrated theorems of Erd\H{o}s-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 $n\ge7$ there are exactly 15 pairwise non-isomorphic such systems (and 13 for $n=6$). We present our result in terms of a hierarchy of Tur\'an numbers $\ex^{(s)}(n, M_2^{3})$, $s\ge1$, 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 $n\ge6$ 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.