Ramsey numbers of cubes versus cliques

Abstract

The cube graph Q n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2n vertices. The Ramsey number r(Q n ;K s ) is the minimum N such that every graph of order N contains the cube graph Q n or an independent set of order s. In 1983, Burr and Erdős asked whether the simple lower bound r(Q n ;K s )≥(s−1)(2n−1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.

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Correspondence to David Conlon.

Additional information

Research supported by a Royal Society University Research Fellowship.

Research supported by a Packard Fellowship, a Simons Fellowship, an MIT NEC Corp. award and NSF grant DMS-1069197.

Research supported in part by a Samsung Scholarship.

Research supported in part by SNSF grant 200021-149111 and by a USA-Israel BSF grant.

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Conlon, D., Fox, J., Lee, C. et al. Ramsey numbers of cubes versus cliques. Combinatorica 36, 37–70 (2016). https://doi.org/10.1007/s00493-014-3010-x

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Mathematics Subject Classification (2000)

  • 05D10
  • 05C55