On Hales-Jewett and Related Numbers

Nathan Conlon; Nayda Farnsworth; Aaron Robertson

On Hales-Jewett and Related Numbers

Combinatorics 2026-07-20 v1

Authors: Nathan Conlon , Nayda Farnsworth , Aaron Robertson

Abstract

We investigate the Hales-Jewett numbers and some variants of them, both computationally and via enumerative and probabilistic arguments. In particular, we give the improved lower bound formula on the combinatorial-geometric variant of the Hales-Jewett numbers. {As a consequence of one of the variants investigated, we show that the Milton Bradley game Connect Four played with 2 players on a 5-dimensional hypercube of any side length can end in a draw.

Keywords

Hales-Jewett numbers combinatorial geometry lower bound Connect Four hypercube

Cite

@article{arxiv.2607.18111,
  title  = {On Hales-Jewett and Related Numbers},
  author = {Nathan Conlon and Nayda Farnsworth and Aaron Robertson},
  journal= {arXiv preprint arXiv:2607.18111},
  year   = {2026}
}