arXiv2026
We introduce a method to embed edge-colored graphs into families of expander graphs, which generalizes a framework developed by Draganić, Krivelevich, and Nenadov (2022). As an application, we show that each family of sufficiently pseudo-random graphs on $n$ vertices contains every edge-colored subdivision of $K_Δ$, provided that the distance between branch vertices in the subdivision is large enough, the average degree of each graph in the family is at least $(1+o(1))Δ$, and the number of vertices in the subdivision is at most $(1-o(1))n$. This extends the subdivision-embedding results of Draganić, Krivelevich, and Nenadov to the colorful setting. Even in the monochromatic case, our work extends the earlier result, which only applied to graphs with average degree at most $O(n^{1/5})$. This work is motivated in part by the problem of finding structures in distance graphs defined over finite vector spaces. In particular, we obtain nearly spanning subdivisions of complete graphs with arbitrary prescribed nonzero edge distances in sufficiently large subsets of these spaces.