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{\bf L\'aszl\'o Lov\'asz}
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{\bf Subgraph Densities in Signed Graphons and the Local Simonovits--Sidorenko Conjecture}
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We prove inequalities between the densities of various bipartite
subgraphs in signed graphs. One of the main inequalities is that the
density of any bipartite graph with girth $2r$ cannot exceed the
density of the $2r$-cycle.

This study is motivated by the Simonovits--Sidorenko conjecture,
which states that the density of a bipartite graph $F$ with $m$ edges
in any graph $G$ is at least the $m$-th power of the edge density of
$G$. Another way of stating this is that the graph $G$ with given
edge density minimizing the number of copies of $F$ is,
asymptotically, a random graph. We prove that this is true locally,
i.e., for graphs $G$ that are ``close'' to a random graph.

Both kinds of results are treated in the framework of graphons
(2-variable functions serving as limit objects for graph sequences),
which in this context was already used by Sidorenko.

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