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In Ellenberg and Gijswijt's groundbreaking work, the authors show that asubset of $\mathbb{Z}_3^{n}$ with no arithmetic progression of length 3 must beof size at most $2.755^n$ (no prior upper bound was known of$(3-\epsilon)^n)$), and provide for any prime $p$ a value $\lambda_p<p$ suchthat any subset of $\mathbb{Z}_p^{n}$ with no arithmetic progression of length3 must be of size at most $\lambda_p^n$.  Blasiak et al showed that the same bounds apply to tri-coloured sum-freesets, which are triples $\{(a_i,b_i,c_i):a_i,b_i,c_i\in\mathbb{Z}_p^{n}\}$ with$a_i+b_j+c_k=0$ if and only if $i=j=k$.  Building on this work, Kleinberg, Sawin and Speyer gave a description of avalue $\mu_p$ such that no tri-coloured sum-free sets of size $e^{\mu_p n}$exist in $\mathbb{Z}_p^{n}$, but for any $\epsilon>0$, such sets of size$e^{(\mu_p-\epsilon) n}$ exist for all sufficiently large $n$. The value of$\mu_p$ was left open, but a conjecture was stated which would imply that$e^{\mu_p}=\lambda_p$, i.e. the Ellenberg-Gijswijt bound is correct for thesum-free set problem.  The purpose of this note is to close that gap. The conjecture of Kleinberg,Sawin and Speyer is true, and the Ellenberg-Gijswijt bound is the correctexponent for the sum-free set problem.

Proof of a Conjecture of Kleinberg-Sawin-Speyer