Dense Edge-Magic Graphs and Thin Additive Bases

TL;DR

This paper investigates the maximum size of sumsets of k-subsets within [n], providing new bounds with implications for additive number theory and graph labelings, including quasi-Sidon sets and edge-magic graphs.

Contribution

It introduces new bounds on s(k,n), linking additive number theory functions to graph labelings and quasi-Sidon sets, advancing understanding in these areas.

Findings

Derived new bounds on s(k,n)

Connected additive number theory to graph labelings

Provided insights into quasi-Sidon sets

Abstract

We study s(k,n), the maximum size of A+A where A is a k-subset of [n]. A few known functions from additive number theory can be expressed via s(k,n). For example, our estimates of s(k,n) imply new bounds on the maximum size of quasi-Sidon sets, a problem posed by Erdos and Freud [J. Number Th.38 (1991) 196-205]. Also, applications to so-called edge-magic labellings of graphs are given.