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Let p be a fixed prime. A triangle in &#120125;np is an ordered triple (x, y, z) of points satisfying x + y + z = 0. Let N = pn = |&#120125;np|. Green proved an arithmetic triangle removal lemma which says that for every ϵ > 0 and prime p, there is a δ > 0 such that if X, Y, Z ⊂ &#120125;np and the number of triangles in X × Y × Z is at most δN2, then we can delete ϵN elements from X, Y, and Z and remove all triangles. Green posed the problem of improving the quantitative bounds on the arithmetic triangle removal lemma, and, in particular, asked whether a polynomial bound holds. Despite considerable attention, prior to this paper, the best known bound, due to the first author, showed that 1/δ can be taken to be an exponential tower of twos of height logarithmic in 1/ϵ. We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in &#120125;np. We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is a computable number Cp such that we may take δ = (ϵ/3)Cp, and we must have δ ≤ ϵCp − 0(1). In particular, C2 = 1 + 1/(5/3 − log2 3) ≈ 13.239, and C3 = 1 + 1/c3 with [EQUATION], and [EQUATION], which gives C3 ≈ 13.901. The proof uses Kleinberg, Sawin, and Speyer's essentially sharp bound on multicolored sum-free sets, which builds on the recent breakthrough on the cap set problem by Croot-Lev-Pach, and the subsequent work by Ellenberg-Gijswijt, Blasiak-Church-Cohn-Grochow-Naslund-Sawin-Umans, and Alon.

A tight bound for Green's arithmetic triangle removal lemma in vector spaces