Ramsey numbers of cubes versus cliques
Author(s)Conlon, David; Fox, Jacob; Lee, Choongbum; Sudakov, Benny
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The cube graph Q[subscript n] is the skeleton of the n-dimensional cube. It is an n-regular graph on 2[superscript n] vertices. The Ramsey number r(Q[subscript n] ;K[subscript s]) is the minimum N such that every graph of order N contains the cube graph Q[subscript n] or an independent set of order s. In 1983, Burr and Erdős asked whether the simple lower bound r(Q[subscript n] ;K[subscript s] )≥(s−1)(2[superscript n] −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.
DepartmentMassachusetts Institute of Technology. Department of Mathematics
Conlon, David, Jacob Fox, Choongbum Lee, and Benny Sudakov. “Ramsey Numbers of Cubes Versus Cliques.” Combinatorica (November 5, 2014).