A hierarchy of maximal intersecting triple systems

We reach beyond the celebrated theorems of Erdös-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 ≥ 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 Turan numbers [formula], s ≥ 1, where [formula] 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 C3 is defined as C3 = {{x1,y3,x2}, {x1,y2,x3}, {x2, y1,x3}}. Along the way we show that the largest intersecting triple system H on n ≥ 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.