Hypergraphs not containing a tight tree with a bounded trunk II: 3-trees with a trunk of size 2

A tight r-tree T is an r-uniform hypergraph that has an edge-ordering e₁,e₂,…,e_t such that for each i≥2, e_i has a vertex v_i that does not belong to any previous edge and e_i−v_i is contained in e_j for some j<i. Kalai conjectured in 1984 that every n-vertex r-uniform hypergraph with more than [Formula presented] edges contains every tight r-tree T with t edges. A trunk T^′ of a tight r-tree T is a tight subtree T^′ of T such that vertices in V(T)∖V(T^′) are leaves in T. Kalai's Conjecture was proved (Frankl and Füredi, 1987) for tight r-trees that have a trunk of size one. In a previous paper (Füredi et al., 2019) we proved an asymptotic version for all tight r-trees that have a trunk of bounded size. In this paper we continue that work to establish the exact form of Kalai's Conjecture for all tight 3-trees of at least 8 edges that have a trunk of size two.