Bounding the size of square-free subgraphs of the hypercube

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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.

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Thomason, A., & Wagner, P. (2009). Bounding the size of square-free subgraphs of the hypercube. Discrete Mathematics, 309(6), 1730–1735.

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