Embeddings of weighted graphs in Erd\H{o}s-type settings

TL;DR

This paper uses graph theory to show that large sets in Erdős-type settings necessarily contain many copies of any given weighted tree, linking set size to the presence of specific weighted configurations.

Contribution

It establishes a lower bound on the number of weighted tree copies in large sets, extending combinatorial results to weighted graph embeddings.

Findings

Large sets contain at least a constant times their size copies of any weighted tree.

The number of copies depends only on the tree and not on the set size.

The method applies graph-theoretic techniques to weighted configurations.

Abstract

Many recent results in combinatorics concern the relationship between the size of a set and the number of distances determined by pairs of points in the set. One extension of this question considers configurations within the set with a specified pattern of distances. In this paper, we use graph-theoretic methods to prove that a sufficiently large set $E$ must contain at least $C_G|E|$ distinct copies of any given weighted tree $G$, where $C_G$ is a constant depending only on the graph $G$.