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arXiv · 2604.01381

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

Alex McDonald
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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.

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Alex McDonald. 2026-04-01. Extremal graph theory and point configurations in Ahlfors-David regular sets. https://arxiv.org/abs/2604.01381

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