A Combinatorial Proof of Kneser’s Conjecture*

Kneser’s conjecture, first proved by Lovász in 1978, states that the graph with all k-element subsets of {1, 2, . . . , n} as vertices and with edges connecting disjoint sets has chromatic number n−2k+2. We derive this result from Tucker’s combinatorial lemma on labeling the vertices of special triangulations of the octahedral ball. By specializing a proof of Tucker’s lemma, we obtain self-contained purely combinatorial proof of Kneser’s conjecture.

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Correspondence to Jiří Matoušek.

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* Research supported by Charles University grants No. 158/99 and 159/99 and by ETH Zürich.

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Matoušek, J. A Combinatorial Proof of Kneser’s Conjecture*. Combinatorica 24, 163–170 (2004). https://doi.org/10.1007/s00493-004-0011-1

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Mathematics Subject Classification (2000):

  • 05C15
  • 05A05
  • 55M35