Determine the constant β(q)

Determine the constant β(q), defined as the eventual constant value of β_{0,q}(n,q+1) for (q+1)-uniform intersecting families, for every integer q≥1.

Background

The paper defines β{p,q}(n,k) as the maximum, over all k-uniform intersecting families on [n], of the minimum number of family members containing a prescribed p-set and avoiding a prescribed disjoint q-set. For each fixed q, the quantity β{0,q}(n,q+1) stabilizes for sufficiently large n; this eventual value is denoted by β(q).

Theorem 1.4 shows that β(q) determines the asymptotic behavior of β_{p,q}(n,k) for every fixed k≥q+1. The authors note that determining β(q) is the principal unresolved problem, and relate it to the undetermined maximum size of a (q+1)-uniform intersecting family with covering number q+1.