A Steiner triple system with a broken chromatic spectrum
Year: 2026
Type: preprint
Abstract: A bicoloring of a Steiner triple system colors its points so that every block receives exactly two colors, and the feasible set of the system is the set of integers $k$ for which some bicoloring uses exactly $k$ colors. Mario Gionfriddo conjectured in 2003 that the feasible set of every bicolorable Steiner triple system has no gaps; Tuza and Voloshin listed the question as open in 2008, and Gionfriddo restated the conjecture in 2025. In 2014 Gionfriddo, Guardo, Milazzo and Voloshin left open whether a system of order 39 has feasible set $\{3,5\}$. We give a Steiner triple system of order 39 that has bicolorings with exactly three and exactly five colors but none with exactly four, so the conjecture is false. With their theorem that bicolorings of systems of order 39 use three, four or five colors, its feasible set is $\{3,5\}$. The class sizes of the two bicolorings come from the tables of Colbourn, Dinitz and Rosa. The absence of a four-color bicoloring is proved by computer: a SAT solver's clausal refutation of a direct encoding, checked by DRAT-trim and by a separate unit-propagation checker. This work has not yet been reviewed by experts. The proofs, the search code, the certificate checkers and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction. Computer-assisted; not yet reviewed by experts. Produced by AI models under the author's direction.
Author: Graham, P.
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225010748