Complete-intersection classification for t-intersecting tree families

Determine an integer j for every pair of parameters t and n such that the family of spanning trees of K_n containing at least t+j edges of a balanced spanning forest F_{n,t+2j} is the largest t-intersecting family of trees.

Background

The paper defines F_{n,ℓ} as a spanning forest whose component sizes are as equal as possible and defines 𝓕{n,t,j} as the family of trees containing at least t+j of the t+2j edges of F{n,t+2j}. This conjecture proposes a complete-intersection-type description of the largest t-intersecting tree family for all parameter values, including regimes in which fixed sets of t disjoint edges are unavailable or non-optimal.