Joonkyung Lee (이준경), The extremal number of subdivisions

One of the cornerstones of extremal graph theory is a result of Füredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if H is a bipartite graph with maximum degree r on one side, then there is a constant C such that every graph with n vertices and C n^{2 – 1/r} edges contains a copy of H. This result is tight up to the constant when H contains a copy of K_r,s with s sufficiently large in terms of r. We conjecture that this is essentially the only situation in which Füredi’s result can be tight and prove this conjecture for r = 2. More precisely, we show that if H is a C₄-free bipartite graph with maximum degree 2 on one side, then there are positive constants C and δ such that every graph with n vertices and C n^{3/2 – δ} edges contains a copy of H. This answers a question by Erdős from 1988. The proof relies on a novel variant of the dependent random choice technique which may be of independent interest. This is joint work with David Conlon.