Nagy Zoltán Lóránt: The Turán number of the blow-ups of trees

A conjecture of Paul Erdős from 1967 asserts that any graph on n vertices which does
not contain a fixed r-degenerate bipartite graph F has at most Cn^(2-1/r) edges, where C is a
constant depending only on F. 

We show that this bound holds for a large family of r-degenerate bipartite graphs, including all blow-ups of trees.  This generalises
many previously proven cases of the Erdős conjecture, including the related results of Füredi
and Alon, Krivelevich and Sudakov. Our proof uses supersaturation and a probabilistic argument in connection with  random walks on
an auxiliary graph.

Joint work with Andrzej Grzesik and Olivér Janzer