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Alexander Sidorenko works on density inequalities in graph theory and the theory of hypergraphs. His name is associated with the Sidorenko conjecture, which compares copies of a fixed bipartite graph with the random-density benchmark.^[ref]

With David Conlon and Joonkyung Lee, he studied hypergraph failures of the analogous inequality. Their result turns such failures into improved extremal constructions, connecting analytic inequalities for densities with concrete forbidden-subgraph problems.^[ref]

David Conlon studies extremal graph theory and Ramsey theory. His joint work with Jacob Fox, Choongbum Lee and Benny Sudakov shows that imposing a vertex order can make Ramsey problems substantially harder, even for matchings.^[ref]

With Joonkyung Lee and Alexander Sidorenko, he also linked failures of a hypergraph analogue of the Sidorenko conjecture to improved extremal lower bounds. This turns a failed density inequality into information about constructions avoiding a specified hypergraph.^[ref]

Joonkyung Lee works on extremal combinatorics. With David Conlon and Alexander Sidorenko, he studied connections between the Sidorenko conjecture and extremal numbers.^[ref]

The graph conjecture concerns lower bounds on homomorphism densities for bipartite graphs. The paper investigates hypergraph analogues, where counterexamples exist, and uses those failures to improve lower bounds for the number of edges in a forbidden-configuration-free hypergraph.^[ref]

The Sidorenko conjecture is the assertion that every finite bipartite graph $H$ satisfiesfor every nonempty finite simple graph $G$. Here $t(H,G)$ is the homomorphism density: the probability that a uniformly random map from $V(H)$ to $V(G)$ sends every edge to an edge. The host graph $G$ need not be bipartite. In the equivalent graphon formulation, the edge constraints are integrated against one symmetric measurable kernel.^[ref]

For the three-vertex graph path, writing $n=|V(G)|$ givesby the Cauchy-Schwarz inequality. The example explains the comparison: correlations between adjacent edge constraints can increase the homomorphism count above the independent-edge heuristic. Bipartiteness is necessary for an assertion covering all hosts, since a nonbipartite $H$ has no homomorphism to a bipartite graph of positive edge density.

Every tree $T$ satisfies the Sidorenko conjecture. If a bipartite host graph has edge density $\alpha$, then a uniformly random bipartition-respecting map from a $k$-vertex tree into the host is a graph homomorphism with probability at least ${\alpha }^{k-1}$.