math.CO — Combinatorics

The constant Steiner kernel is not optimal at large girth

Contributed by Matthew Kwan, Huy Tuan Pham

In the paper "nonlinear lower-tail large deviations at criticality", we bound the number of order-$n$ Steiner triple systems with no Pasch configuration. In doing so, we show that a certain variational problem is minimised by a constant kernel. Using ChatGPT 6 Pro, we also discovered that the analogous variational problems for Steiner triple systems of girth at least $g$, for large $g$, are *not* minimised by the constant kernel, and we included a brief remark to this effect. In this note (not intended for publication), we present the details of this proof.