On the number of graphs without 4-cycles

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Abstract

The number Fn of n-vertex graphs lacking a 4-cycle is shown to satisfy 12n32⩽log Fn⩽ 1.08192n32 for large n. This represents an improvement of the previously known upper and lower bounds and resolves a long-standing question posed by Erdös. The result is obtained by using an argument previously used by the authors to bound the numbers of lattices on n elements.

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Supported in part by ONR Contract N00014-76-C-0366