Sets of integers avoiding congruent subsets

TL;DR

This paper investigates sets of integers that avoid certain congruent subsets, refining previous results by Erdős and Harzheim on the structure and properties of such sets.

Contribution

It provides improved bounds and new insights into the structure of integer sets avoiding congruent subsets, advancing the understanding of their combinatorial properties.

Findings

Refined bounds on the size of sets avoiding congruent subsets

New structural characterizations of such sets

Enhanced understanding of their combinatorial properties

Abstract

Consider the sets of integers $A$ that avoid any arrangement of $g$ congruent $h$-subsets. Our findings refine and improve upon some results by Erd\H{o}s and Harzheim about these sets.