Sharp finite bound for intersecting families without small transversals

Prove or refute the conjecture that, for every positive integer \(n\), the maximal size \(t(n)\) of the union of a family \(\mathcal{F}\subseteq[\omega]^n\) having the intersection property and no transversal of size \(n-1\) is \((n-1)^2+n\).

Background

The paper introduces a finite-combinatorial quantity t(n)t(n) for intersecting families of nn-element sets that do not admit a transversal of size n−1n-1. Projective planes of order n−1n-1 give lower-bound examples of size (n−1)2+n(n-1)^2+n. The authors explicitly conjecture that this lower bound is optimal for every nn.

References

In fact, we conjecture that $t(n) = (n-1)2 + n$ for every $n$.

— Non-separable combinatorial Banach spaces  (2610.01919 - Borodulin-Nadzieja et al., 1 Oct 2026) in Section 8, Trees and geometry of balls, Remark following Proposition 19