A Salem-Spencer-Type Construction for Large Subsets of Integer Grids with No Isosceles Right Triangles

2026/07/24 by Gyula Károlyi, Jozsef Solymosi
Mathematics · #math.CO #math.NT

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Abstract

Let F(n) be the largest size of a subset of \0,1,…,n-1\² containing no nondegenerate isosceles right triangle. We give a modified Salem--Spencer-type construction over the Gaussian integers showing that F(n)=Ω(n^1.3). The best known upper bound is F(n)≪ n²/(log n)^1+c for some absolute constant c>0, so there is still a large gap between the bounds.

Citations

• AlphaEvolve: A coding agent for scientific and algorithmic discovery
• An improvement to the Kelley-Meka bounds on three-term arithmetic progressions
• Strong Bounds for 3-Progressions

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