Extremal graph theory and point configurations in Ahlfors-David regular sets

Abstract

We study the problem of embedding bipartite graphs in Ahlfors-David regular sets of large dimension using results from extremal graph theory. Our main theorem states that any graph satisfying a power-improving bound on the extremal number can be found in the distance graph of a sufficiently high-dimensional AD-regular set. In particular, we show that AD-regular sets of dimension greater than $\frac{d+1}{2}$ must contain even cycles of all lengths if $d\ge 3$, and must contain even cycles of length at least 6 if $d=2$. This improves the best known threshold for the problem in $d\ge 4$, and yields entirely new results in $d=2,3$, under the extra assumption of AD-regularity. We also prove analogous results for large subsets of vector spaces over finite fields, which improve the best known exponent for even cycles in all dimensions.