Embedding bipartite distance graphs under Hamming metric in finite fields

Abstract

The famous Erdős-Falconer distance problems aim to quantify the extent to which large sets must determine many distinct distances. The problems in both the Euclidean setting and the discrete setting have received much attention. An interesting generalization of Erdős-Falconer problems is to find locating point configurations with prescribed metric structure within large subsets. In this paper, we consider such problem in the discrete setting. We say a distance graph H in Fqn under Hamming metric means a graph H=(V,E) with vertices as the points in V⊆Fqn and edges being assigned by Hamming distance between the pairs of vertices. We call a subset A⊆Fqn contains an isometric copy of distance graph H, if there exists an embedding ϕ:V(H)→A such that for every edge uv∈E(H), the Hamming distance between ϕ(u) and ϕ(v) is equal to the Hamming distance between u and v. When q is a fixed prime power and n goes to infinity, we show that for any given bipartite graph H with ex(n,H)=O(n), any subset A⊆Fqn contains a positive proportion of all possible isometric copies of distance graphs H whose edges are assigned by the same Hamming distance, if |A|>q(1−c)n for some c=c(q,H)>0. Our methods also work for general bipartite graphs. Moreover, we will give the quantitative bounds for the above constant c=c(q,H).Unlike using analytical method as before, our ideas are more combinatorial, which mainly include two techniques. The first one is the celebrated dependent random choice, which is like glue, sticking several small structures together. The second one is a new extension of the modular version of the classical Delsarte's inequality. In particular, this extension itself is of independent interest, which also helps us build a surprising connection between our main results and the extremal number of the bipartite graphs.