Bounding the size of square-free subgraphs of the hypercube

Abstract

We investigate the maximum size of a subset of the edges of the n-cube that does not contain a square, or 4-cycle. The size of such a subset is trivially at most 3/4 of the total number of edges, but the proportion was conjectured by Erdo{double acute}s to be asymptotically 1/2. Following a computer investigation of the 4-cube and the 5-cube, we improve the known upper bound from 0.62284... to 0.62256... in the limit. © 2008 Elsevier B.V. All rights reserved.