[Paper Review] Square-Difference-Free Sets of Size Omega(n^{0.7334...})

Richard Beigel, William Gasarch|ArXiv.org|Apr 30, 2008

Limits and Structures in Graph Theory8 references3 citations

TL;DR

This paper improves the lower bound for the size of the largest square-difference-free subset of {1, ..., n}, showing it is Ω(n^{0.7334...}), by constructing such sets using modular arithmetic and recursive composition over a squarefree modulus m = 205 with a carefully chosen subset S of size 12. The result strengthens Ruzsa's earlier bound and yields a new lower bound on quadratic van der Waerden numbers.

ABSTRACT

A set A is square-difference free (henceforth SDF) if there do not exist x,y\in A, x e y, such that |x-y| is a square. Let sdf(n) be the size of the largest SDF subset of {1,...,n}. Ruzsa has shown that sdf(n) = Ω(n^{0.5(1+ \log_{65} 7)}) = Ω(n^{0.733077...}) We improve on the lower bound by showing sdf(n) = Ω(n^{0.5(1+ \log_{205} 12)})= Ω(n^{.7443...}) As a corollary we obtain a new lower bound on the quadratic van der Waerden numbers.

Motivation & Objective

• To improve the known lower bound on the size of the largest square-difference-free subset of {1, ..., n}, denoted sdf(n).
• To extend Ruzsa's method of constructing square-difference-free sets modulo m using recursive composition and modular arithmetic.
• To provide a new lower bound for sdf(n) that exceeds Ruzsa's previous result of Ω(n^{0.733077...}).
• To derive a new lower bound on the quadratic van der Waerden numbers via the improved sdf(n) bound.
• To identify the open problem of closing the gap between the current lower bound and the known upper bound of O(n / log^{c_n} n) with c_n → ∞.

Proposed method

• Constructs square-difference-free sets modulo m by selecting a squarefree modulus m and a subset S ⊆ [m] such that no two elements in S differ by a square modulo m.
• Uses recursive composition: given an SDF set modulo m and another modulo m^{2k-2}, constructs a larger SDF set modulo m^{2k} via linear combinations.
• Applies Lemma 3.5 to derive a recurrence: sdfmod(m^{2k}) ≥ m · sdfmod(m) · sdfmod(m^{2k-2}), enabling exponential growth in set size.
• Employs Lemma 3.6 to relate the growth rate of sdfmod(n) to the logarithmic ratio log_m |S|, yielding a lower bound of Ω(n^{0.5(1 + log_m |S|)}).
• Selects m = 205 and a 12-element subset S of residues modulo 205 with no square differences, achieving log_{205} 12 > 0.4668.
• Combines the modular construction with a covering argument using translates to derive a lower bound on the function f(c) in square-difference-free colorings.

Experimental results

Research questions

• RQ1 Can the lower bound on the size of the largest square-difference-free subset of {1, ..., n} be improved beyond Ruzsa's Ω(n^{0.733077...})?
• RQ2 What is the maximal value of log_m |S| for a squarefree modulus m and a subset S ⊆ [m] with no two elements differing by a square modulo m?
• RQ3 How does the improved lower bound on sdf(n) affect the known bounds for quadratic van der Waerden numbers?
• RQ4 Can the gap between the current lower bound Ω(n^{0.7334...}) and the upper bound O(n / log^{c_n} n) be closed?
• RQ5 What modular constructions yield the largest possible growth rate for square-difference-free sets via recursive composition?

Key findings

• The paper establishes a new lower bound: sdf(n) = Ω(n^{0.7334...}), improving upon Ruzsa's previous bound of Ω(n^{0.733077...}).
• The improvement is achieved by constructing an SDF set modulo m = 205 using a 12-element subset S ⊆ {0, 2, 8, 14, 77, 79, 85, 96, 103, 109, 111, 181} with no square differences modulo 205.
• The growth rate is derived via Lemma 3.6, showing sdf(n) ≥ Ω(n^{0.5(1 + log_{205} 12)}), where log_{205} 12 ≈ 0.4668, yielding an exponent of 0.7334.
• The method generalizes Ruzsa's approach by using a larger modulus and a larger subset S, resulting in a higher exponent.
• As a corollary, the paper derives a new lower bound f(c) ≥ Ω(c^{3.75}) for the function f(c) in square-difference-free colorings, where f(c) is the minimal N such that any c-coloring of [N] contains a monochromatic pair differing by a square.
• The paper leaves open the problem of closing the gap between the current lower bound Ω(n^{0.7334...}) and the upper bound O(n / log^{c_n} n) with c_n → ∞.