Spectral complete-intersection conjecture for t-intersecting uniform families

Establish that, for every pair of integers satisfying 1≤t≤k and n>2k−t, at least one Frankl family _r={F∈([n] choose k): |F∩[t+2r]|≥t+r}, with 0≤r≤k−t, has spectral radius equal to the maximum adjacency-tensor spectral radius among all t-intersecting subfamilies of ([n] choose k); equivalently, prove that every such family satisfies ρ()≤max_{0≤r≤k−t}ρ(_r).

Background

The paper studies the adjacency-tensor spectral radius of t-intersecting k-uniform families and proves the desired Frankl-family extremality in substantial regimes. For t≥2, it resolves the value problem when the full t-star has spectral radius at least that of the first nontrivial Frankl family, and for t=1 it determines the extremal value throughout n≥2k, with separate endpoint behavior at n=2k.

The unresolved general problem is whether the complete-intersection phenomenon persists spectrally outside the star-dominant range: namely, whether the maximum over all t-intersecting families is always attained by one of the Frankl families. The paper explicitly distinguishes this value conjecture from the stronger and potentially separate equality classification asserting that a uniquely maximizing Frankl family is the unique extremal structure up to permutation.