Largest Sidon subsets in weak Sidon sets

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 $$ g(n) := \min{\, h(A) : A \subset \mathbb{R},\ |A| = n,\ A \text{ is a weak Sidon set} \,}. $$ Sárközy and Sós 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 proving that $$ \lim_{n\to\infty} \frac{g(n)}{n} = \frac{1}{2}. $$ We also investigate a related problem of Erdős 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. Erdős 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 $. Gyárfás 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 $c_*$.