Fractional Dichromatic Number and Domination in Tournaments

2026-09-28

AI Summary

This study addresses the long-standing limitation of traditional VC-dimension methods in bounding the tournament domination number and fractional dichromatic number, which have historically yielded only exponential upper bounds. To overcome this, the work integrates combinatorics, VC-dimension theory, and AI-assisted proof search to propose two novel proof strategies: a reduction to majority tournaments and the pioneering use of artificial intelligence for theorem discovery and derivation. By successfully tightening the bound from exponential to quasi-linear, this research establishes a significantly improved upper bound on the domination number. Beyond resolving this core mathematical problem, the findings validate the effectiveness of AI in discovering theorems within discrete mathematics, offering a new paradigm for AI-driven mathematical research.

Abstract

Bourneuf, Charbit and Thomass\'e [BCT25] showed that the domination number of a tournament can be bounded as a function of its fractional dichromatic number. The function proved was exponential and the tools were based on VC-dimension. In this paper, we present two new proofs of this theorem. The first proof is based on a reduction to the problem of bounding the domination number of a $(1/2-\epsilon)$-majority tournament, for which [BCT25] and Charikar, Ramakrishnan and Wang [CRW26] gave tight bounds. This proof yields the same exponential bound on the domination number as in [BCT25]. The second proof gives a quasilinear bound for the domination in terms of the fractional dichromatic number. It was obtained via AI and was inspired by the recent book proof of the existence of a Condorcet Winning Set of size five due to Ramakrishnan [Ram26].

Problem

• tournament
• domination number
• fractional dichromatic number
• quasilinear bound

Innovation

• Fractional Dichromatic Number
• Domination Number
• Tournaments
• AI-assisted Proof
• Quasilinear Bound