Balanced clique subdivisions and cycles lengths in Kₒ, ₓ-free graphs

Let t s2 be integers. Confirming a conjecture of Mader, Liu and Montgomery J. Lond. Math. Soc. , 2017 showed that every Kₒ, ₓ-free graph with average degree d contains a subdivision of a clique with at least (d^s{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, Erdos 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 (12 -od (1) ) d. In this paper, we improve this low bound to (s2 (s-1) -od (1) ) d for Kₒ, ₓ-free graphs. Both proofs of our results use the graph sublinear expansion property as well as some novel structural techniques.