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

We solve Green's problem, proving an essentially tight bound for Green's arithmetic triangle removal lemma in Fpn. We show that a polynomial bound holds, and further determine the best possible exponent. Namely, there is an explicit number Cp such that we may take δ=(ϵ/3)Cp, and we must have δ≤ϵCp−o(1). In particular, C2=1+1/(5/3−log2⁡3)≈13.239, and C3=1+1/c3 with c3=1−log⁡blog⁡3, b=a−2/3+a1/3+a4/3, and a=33−18, which gives C3≈13.901. The proof uses the essentially sharp bound on multicolored sum-free sets due to work of Kleinberg-Sawin-Speyer, Norin, and Pebody, 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.