Abstract
The Turán number of a graph H, denoted ex(n,H), is the maximum number of edges in an n-vertex graph with no subgraph isomorphic to H. Solymosi (2011) conjectured that if H is any graph and ex(n,H)=O(nα) where α>1, then any n-vertex graph with the property that each edge lies in exactly one copy of H has o(nα) edges. This can be viewed as conjecturing a possible extension of the removal lemma to sparse graphs, and is well-known to be true when H is a non-bipartite graph, in particular when H is a triangle, due to Ruzsa and Szemerédi (1978). Using Sidon sets we exhibit infinitely many bipartite graphs H for which the conjecture is false.
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