Tightness of the weighted Hoffman bound for downsets

Determine whether the weighted Hoffman bound is tight for the Kneser graph of every finite downset; equivalently, establish that the weighted Hoffman bound equals the maximum size of an intersecting family in every finite downset.

Background

For a finite downset $\mathcal{D}$, the paper considers the Kneser graph $\mathrm{Kn}(\mathcal{D})$, whose vertices are the members of $\mathcal{D}$ and whose edges join disjoint sets. Independent sets in this graph are exactly intersecting families in $\mathcal{D}$.

The weighted Hoffman bound provides an upper bound on the independence number using a suitably normalized real symmetric matrix supported on the graph. Extensive experiments on thousands of structured and random downsets indicated that this bound always equals the size of a largest star, which is the maximum size of an intersecting family by the paper’s proved strengthening of Chvátal’s conjecture. The authors state that existing proofs do not produce a matrix certifying tightness of the weighted Hoffman bound.