Work

Undecidability of Polynomial Inequalities in Tournaments

Year: 2025

Type: article

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 Lovász. 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.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnaf131

Authors: Hao Chen, Yupeng Lin, Jie Ma, Fan Wei

Institutions: University of Science and Technology of China, Duke University

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: green

Funders: National Science Foundation, National Natural Science Foundation of China, National Key Research and Development Program of China

Awards: G1330075977, Extremal Combinatorics: Themes and Challenging Problems, On Regularity Methods and Applications in Graph Theory