Extremal connectivity for topological cliques in bipartite graphs

Let d ( s ) be the smallest number such that every graph of average degree > d ( s ) contains a subdivision of K s . So far, the best known asymptotic bounds for d ( s ) are ( 1 + o ( 1 ) ) 9 s 2 / 64 ⩽ d ( s ) ⩽ ( 1 + o ( 1 ) ) s 2 / 2 . As observed by Łuczak, the lower bound is obtained by considering bipartite random graphs. Since with high probability the connectivity of these random graphs is about the same as their average degree, a connectivity of ( 1 + o ( 1 ) ) 9 s 2 / 64 is necessary to guarantee a subdivided K s . Our main result shows that for bipartite graphs this gives the correct asymptotics. We also prove that in the non-bipartite case a connectivity of ( 1 + o ( 1 ) ) s 2 / 4 suffices to force a subdivision of K s . Moreover, we slightly improve the constant in the upper bound for d ( s ) from 1 / 2 (which is due to Komlós and Szemerédi) to 10 / 23 .