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

TL;DR

This paper connects extremal graph theory with geometric measure theory to identify conditions under which certain bipartite graphs, including even cycles, must appear in high-dimensional Ahlfors-David regular sets and finite field subsets.

Contribution

It establishes new thresholds for the existence of even cycles in AD-regular sets of various dimensions, improving previous bounds and applying extremal graph theory to geometric configurations.

Findings

AD-regular sets of dimension > (d+1)/2 contain all even cycles for d ≥ 3

Such sets contain even cycles of length ≥ 6 when d=2

Results extend to subsets of finite vector spaces, improving bounds for even cycles

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 $d\geq 3$, and must contain even cycles of length at least 6 if $d=2$. This improves the best known threshold for the problem in $d\geq 4$, 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.