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Undecidability of polynomial inequalities in tournaments

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arxiv 2412.04972 v2 pith:6CLJTWBW submitted 2024-12-06 math.CO

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keywords inequalitiespolynomialdensitieshomomorphismundecidablegraphproblemresult
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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.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. On Domination Exponents for Pairs of Graphs

    math.CO 2025-06 accept novelty 7.0 of 10

    Exact homomorphism density domination exponents are determined for all path pairs and for even cycles against Hamiltonian-cycle graphs, with asymptotically sharp bounds for odd cycles.

  2. Undecidability of Polynomial Inequalities in Subset Densities and Additive Energies

    math.CO 2025-05 conditional novelty 7.0 of 10

    Deciding whether an arbitrary integer polynomial in subset densities and additive energies is nonnegative for all subsets of all finite abelian groups is undecidable.

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