BALANCED SUBDIVISIONS OF A LARGE CLIQUE IN GRAPHS WITH HIGH AVERAGE DEGREE*

In 1984, Thomassen conjectured that for every constant k \in N, there exists d such that every graph with average degree at least d contains a balanced subdivision of a complete graph on k vertices, i.e., a subdivision in which each edge is subdivided the same number of times. Recently, Liu and Montgomery confirmed Thomassen's conjecture. We show that for every constant 0 < c < 1/2, every graph with average degree at least d contains a balanced subdivision of a complete graph of size at least \Omega (dc). Note that this bound is almost optimal. Moreover, we show that every sparse expander with minimum degree at least d contains a balanced subdivision of a complete graph of size at least \Omega (d).