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Largest Sidon subsets in weak Sidon sets

Jie Ma, Quanyu Tang

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 $c_*$.

Largest Sidon subsets in weak Sidon sets

Abstract

A finite set is called a Sidon set if all sums with and are distinct, and a weak Sidon set if all sums with and are distinct. For a finite set , let denote the maximum size of a Sidon subset of , and define Sárközy and Sós asked whether the limit exists and, if so, to determine its value. We resolve this problem completely by proving that We also investigate a related problem of Erdős concerning a local difference condition. A finite set is called a -set if every -element subset of determines at least five distinct values among its six pairwise absolute differences. Erdős asked for the optimal constant such that every -set of size contains a Sidon subset of size at least . Gyárfás and Lehel reduced this to an extremal problem of -uniform hypergraphs and proved . We improve both bounds by establishing 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 .
Paper Structure (18 sections, 25 theorems, 64 equations, 2 figures)

This paper contains 18 sections, 25 theorems, 64 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A problem of Sarkozy and Sos on weak Sidon sets
  3. A problem of Erdos on (4,5)-sets
  4. Proof strategy
  5. Paper organization
  6. Preliminaries
  7. Sidon sets, (4,5)-sets, and weak Sidon sets
  8. A.P.-hypergraphs
  9. Existence of the asymptotic constants
  10. Weak Sidon sets: existence of the limit
  11. (4,5)-sets: existence of the limit
  12. Sidon subsets in weak Sidon sets
  13. Upper bound on gamma*
  14. Lower bound on gamma*
  15. Sidon subsets in (4,5)-sets
  16. ...and 3 more sections

Key Result

Theorem 1.2

The limit $\gamma_* := \lim_{n\to\infty} \frac{g(n)}{n}$ exists and satisfies $\lim_{n\to\infty} \frac{g(n)}{n} = \inf_{n\ge 1} \frac{g(n)}{n}.$

Figures (2)

  • Figure 1: The $7$-vertex $3$-uniform hypergraph $F_7$.
  • Figure 2: Inclusions among Sidon sets, $(4,5)$-sets, and weak Sidon sets.

Theorems & Definitions (49)

  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.5
  • Theorem 1.6
  • Definition 1.7: A.P.-hypergraph
  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • ...and 39 more