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The constant Steiner kernel is not optimal at large girth

Primarily AI-generated textHuman understanding: all partsmath.CO — Combinatorics

Contributed by Matthew Kwan ↗, Huy Tuan Pham ↗

Version 1 / Nov 06, 2026 / CC BY 4.0

Abstract

In the paper "nonlinear lower-tail large deviations at criticality", we bound the number of order-nn 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 gg, for large gg, 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.

Tools used

Anthropic
Claude OpusVersion 5.5
OpenAI
ChatGPTVersion 6 Pro

References

  1. M. Delcourt and L. Postle, Proof of the high girth existence conjecture via refined absorption , arXiv preprint arXiv:2402.17856 (2024).arXiv
  2. S. Glock, D. Kühn, A. Lo, and D. Osthus, On a conjecture of E rd ő s on locally sparse S teiner triple systems , Combinatorica 40 (2020), no. 3, 363--403.
  3. M. Kwan and H. T. Pham, Nonlinear lower-tail large deviations at criticality , arXiv preprint arXiv:2610.10233 (2026).arXiv
  4. M. Kwan, A. Sah, M. Sawhney, and M. Simkin, High-girth S teiner triple systems , Ann. of Math. (2) 200 (2024), no. 3, 1059--1156.
  5. A. Sidorenko, On positive hypergraphs , European J. Combin. 106 (2022), Paper No. 103574, 5.
  6. N. C. Wormald, Models of random regular graphs , Surveys in combinatorics, 1999 ( C anterbury), London Math. Soc. Lecture Note Ser., vol. 267, Cambridge Univ. Press, Cambridge, 1999, pp. 239--298.

Version history

  1. v1 · 7 pagesSubmitted by Matthew KwanInitial depositCurrentNov 06, 2026