Balanced clique subdivisions and cycles lengths in Ks, t-free graphs

Let t≥ s≥2 be integers. Confirming a conjecture of Mader, Liu and Montgomery [J. Lond. Math. Soc., 2017] showed that every Ks, t-free graph with average degree d contains a subdivision of a clique with at least Ω(ds/2(s-1)) vertices. We give an improvement by showing that such a graph contains a balanced subdivision of a clique with the same order, where a balanced subdivision is a subdivision in which each edge is subdivided the same number of times. In 1975, Erd{o}s asked whether the sum of the reciprocals of the cycle lengths in a graph with infinite average degree d is necessarily infinite. Recently, Liu and Montgomery [J. Amer. Math. Soc., 2023] confirmed the asymptotically correct lower bound on the reciprocals of the cycle lengths, and provided a lower bound of at least (1/2 -od(1)) log d. In this paper, we improve this low bound to (s/2(s-1) -od(1)) log d for Ks, t-free graphs. Both proofs of our results use the graph sublinear expansion property as well as some novel structural techniques.