Large sets avoiding x, x+y and x+y²

How large can A⊆{1,…,N} be if it contains no configuration {x,x+y,x+y²} for a nonzero integer y?

Why it matters and what counts as progress

Why it matters: Explicit lower bounds test how sparse a set must become to avoid this nonlinear additive pattern.

What counts as progress: A construction improving the exponent in |A|≥N^α, with its finite residue conditions verified and the treatment of negative y retained.

How the problem developed

16 Aug 2019 · Lower bound construction [1]
Khalid Younis Constructed configuration-free subsets of size on the order of N^0.7685 for the pattern {x,x+y,x+y²}.
11 Sept 2026 · Formalized stronger instance [2]
JD-Jones-ASES, with Claude Fable 5.1 credited in the repository commit Released a Lean formalization of a residue-set instance at modulus 145 with exponent 0.770287…, exceeding Younis’s cited modulus-65 exponent by exact comparison.

AI contribution

AI-found residue sets improve a polynomial-free lower bound

For positive-integer sets avoiding {x,x+y,x+y²} for every nonzero integer y, Claude-found residue sets modulo 145 of sizes 10 and 32 satisfy Younis’s two-set conditions. They give lower-bound exponent 1/2+log(10)/(3log(145))+log(32)/(6log(145)), between 0.7702 and 0.77029.

AI involvementAI-led

ImportanceNotable

Technical verificationFormal (Lean 4/Mathlib kernel with ordinary decide and transitive axiom audit; independent exact Python enumeration of finite data)

Human understandingFew