Undecidability of polynomial inequalities in tournaments

TL;DR

This paper proves that determining the validity of polynomial inequalities in tournament homomorphism densities is undecidable, extending previous results from undirected graphs and weighted graphs to directed tournaments.

Contribution

It establishes the undecidability of polynomial inequalities in tournament homomorphism densities, filling a gap in the understanding of complexity in directed graph settings.

Findings

Undecidability of polynomial inequalities in tournaments proven

Extends previous undecidability results to directed tournaments

Supports the broader conjecture of computational complexity in graph homomorphism problems

Abstract

Many fundamental problems in extremal combinatorics are equivalent to proving certain polynomial inequalities in graph homomorphism densities. In 2011, a breakthrough result by Hatami and Norine showed that it is undecidable to verify polynomial inequalities in graph homomorphism densities. Recently, Blekherman, Raymond and Wei extended this result by showing that it is also undecidable to determine the validity of polynomial inequalities in homomorphism densities for weighted graphs with edge weights taking real values. These two results resolved a question of Lov\'asz. In this paper, we consider the problem of determining the validity of polynomial inequalities in digraph homomorphism densities for tournaments. We prove that the answer to this problem is also undecidable.