The Graph-codes

The symmetric difference of two graphs G(1), G(2) on the same set of vertices [n] = {1, 2, ... , n} is the graph on [n] whose set of edges are all edges that belong to exactly one of the two graphs G(1), G(2). Let H be a fixed graph with an even (positive) number of edges, and let D-H(n) denote the maximum possible cardinality of a family of graphs on [n] containing no two members whose symmetric difference is a copy of H. Is it true that D-H(n) = o(2((n)(2))) for any such H? We discuss this problem, compute the value of D-H(n) up to a constant factor for stars and matchings, and discuss several variants of the problem including ones that have been considered in earlier work.(c) 2023 Published by Elsevier Ltd.