The Chromatic Number of Kneser Hypergraphs via Consensus Division

Author Ishay Haviv

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Ishay Haviv
  • School of Computer Science, The Academic College of Tel Aviv-Yaffo, Israel


We are grateful to Aris Filos-Ratsikas for a fruitful discussion and for a clarification on [Argyrios Deligkas et al., 2022] and to the anonymous reviewers for their useful comments.

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Ishay Haviv. The Chromatic Number of Kneser Hypergraphs via Consensus Division. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 60:1-60:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)


We show that the Consensus Division theorem implies lower bounds on the chromatic number of Kneser hypergraphs, offering a novel proof for a result of Alon, Frankl, and Lovász (Trans. Amer. Math. Soc., 1986) and for its generalization by Kriz (Trans. Amer. Math. Soc., 1992). Our approach is applied to study the computational complexity of the total search problem Kneser^p, which given a succinct representation of a coloring of a p-uniform Kneser hypergraph with fewer colors than its chromatic number, asks to find a monochromatic hyperedge. We prove that for every prime p, the Kneser^p problem with an extended access to the input coloring is efficiently reducible to a quite weak approximation of the Consensus Division problem with p shares. In particular, for p = 2, the problem is efficiently reducible to any non-trivial approximation of the Consensus Halving problem on normalized monotone functions. We further show that for every prime p, the Kneser^p problem lies in the complexity class PPA-p. As an application, we establish limitations on the complexity of the Kneser^p problem, restricted to colorings with a bounded number of colors.

Subject Classification

ACM Subject Classification
  • Theory of computation → Problems, reductions and completeness
  • Mathematics of computing → Graph theory
  • Kneser hypergraphs
  • consensus division
  • the complexity classes PPA-p


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