Largest Sidon subsets in weak Sidon sets

TL;DR

This paper determines the exact maximum size of Sidon subsets within weak Sidon sets, resolving a longstanding question, and also improves bounds on a related problem involving local difference conditions in finite sets.

Contribution

It provides an exact formula for the function g(n) related to Sidon subsets in weak Sidon sets and refines bounds on the constant c_* for sets with specific difference properties.

Findings

g(n) = ⌈(n+1)/2⌉ for all n ≥ 1

Limit of g(n)/n as n→∞ is 1/2

Improved bounds for c_*: 9/17 ≤ c_* ≤ 4/7

Abstract

A finite set $ S \subset \mathbb{R} $ is called a Sidon set if all sums $ x+y $ with $ x,y \in S $ and $ x \le y $ are distinct, and a weak Sidon set if all sums $ x+y $ with $ x,y \in S $ and $ x < y $ are distinct. For a finite set $ A \subset \mathbb{R} $, let $ h(A) $ denote the maximum size of a Sidon subset of $ A $, and define $$ g(n) := \min\{\, h(A) : A \subset \mathbb{R},\ |A| = n,\ A \text{ is a weak Sidon set} \,\}. $$ S\'ark\"ozy and S\'os asked whether the limit $ \lim_{n\to\infty} g(n)/n $ exists and, if so, to determine its value. We resolve this problem completely by determining $g(n)$ exactly: $$ g(n)=\left\lceil \frac{n+1}{2}\right\rceil \qquad\text{for all } n\ge 1. $$ In particular, $\lim_{n\to\infty} g(n)/n=\frac12$. We also investigate a related problem of Erd\H{o}s concerning a local difference condition. A finite set $ A \subset \mathbb{R} $ is called a $(4,5)$-set if every $4$-element subset of $A$ determines at least five distinct values among its six pairwise absolute differences. Erd\H{o}s asked for the optimal constant $ c_* > 0 $ such that every $(4,5)$-set of size $ n $ contains a Sidon subset of size at least $ c_* n $. Gy\'arf\'as and Lehel reduced this to an extremal problem of $3$-uniform hypergraphs and proved $\frac{1}{2} + \frac{1}{141 \cdot 76} \le c_* \le \frac{3}{5}$. We improve both bounds by establishing $$ \frac{9}{17} \le c_* \le \frac{4}{7}, $$ where the lower bound uses a reformulation of the extremal problem, and the upper bound follows from an explicit construction together with a convenient characterization of $c_*$.