More two-distance counterexamples to Borsuk's conjecture from strongly regular graphs

TL;DR

This paper constructs new counterexamples to Borsuk's conjecture using strongly regular graphs, extending previous work by identifying additional graphs that produce high-dimensional counterexamples.

Contribution

It introduces two new strongly regular graphs that generate counterexamples to Borsuk's conjecture in various high dimensions, expanding the set of known counterexamples.

Findings

Counterexamples in dimensions 764 to 781 derived from new graphs.

Counterexample in 240 dimensions from a smaller graph.

Use of computational methods within GAP for extensive calculations.

Abstract

In 2013 Andriy V. Bondarenko showed how to construct a two-distance counterexample to Borsuk's conjecture from any strongly regular graph whose vertex set is not the union of at most $f+1$ cliques (sets of pairwise adjacent vertices) where $f$ is the multiplicity of the second-largest eigenvalue of its adjacency matrix. He applied that construction to those two graphs that he had been able to prove to fulfill the condition: From the $G_2(4)$ graph (on 416 vertices) he got a 65-dimensional two-distance counterexample. From the $Fi_{23}$ graph (on 31671 vertices) he got a 782-dimensional one and, by considering certain induced subgraphs, counterexamples in dimensions 781, 780 and 779. This article presents two other strongly regular graphs fulfilling the condition, on 28431 and on 2401, resp., vertices. It gives dedicated counterexamples in dimensions from 781 down to 764 derived from the bigger graph (that turned out to be an induced subgraph of the $Fi_{23}$ graph) and a 240-dimensional counterexample derived from the smaller graph. Several contained propositions rely on the results of (often extensive) computations, mainly within the computer algebra system GAP. The source package contains (almost) all used source files.