Extremal affine copies of {0,1,3}

• Slug: affine-013
• List: P05
• Solver: SuperGrok CLI grok-4.6 --reasoning-effort xhigh
• Status: open
• Area: Extremal additive combinatorics
• Sources: Green 100 #24; Aaronson arXiv:1805.01980
• Started: 2026-08-17

Statement

For an n-element integer set A, the conjectured maximum number of affine copies of {0,1,3} is (1/3+o(1))n^2, but even the asymptotic constant is unknown.

Tonight

A certified small-n extremal table, a construction beating the published constant on infinitely many n, or an exact upper bound that moves the constant. Isolated small-n counts are an incomplete search unless they imply a new infinite-family bound. Fetch Green #24 before searching.

Result (2026-08-17)

Aaronson’s T(S) (ordered triples with x+2y=3z) satisfies T(S) ≤ ⌈n²/2⌉ for every n-element S ⊂ ℤ, by an ordering-and-injection argument adapted from Green–Sisask’s 3AP count. Hence γ_{1,2,-3} ≤ 1/2, which is strictly below the published Hardy–Littlewood / Aaronson 3/4. Constructions tonight do not beat the interval’s 1/3 (the almost-interval {0,…,n−2,n} for n=3m has T = n²/3+1; a sporadic 7-set has T=18). The conjecture γ=1/3 remains open. Verifier: compute/verify_half.py.

Result (2026-08-27)

Residue. Endpoint induction and a second family search in compute/q1/ do not move 1/2. The 2/3 end-budget that would give γ=1/3 already fails at n=5; both ends can score 9/10 of n−1 without beating the interval. No certified family has limsup T/n² > 1/3. Replay: compute/q1/run_all.sh. Cert: compute/q1/certs/q1.json.