The Six-Dimensional Tarski Problem is $Ω(\log^3 n)$-Hard

Abstract: We study the deterministic query complexity of finding a fixed point of a monotone function $f:[n]<sup>d\to[n]<sup>d$, where querying $x$ returns $f(x)$. For fixed $d\ge2$, earlier work established a lower bound of $Ω((\log n)<sup>2)$ and an upper bound of $O_d((\log n)<sup>{\lceil(d+2)/3\rceil})$ for the query complexity. Very recently, several papers obtained improved lower bounds using AI-assisted proofs. In this paper, we further improve these lower bounds to $$\text{Tarski}(n,d)=Ω_d((\log n)<sup>{2+\lfloor\log_3(d/2)\rfloor})$$. In particular, we obtain the optimal $Θ((\log n)<sup>3)$ deterministic query complexity in six dimensions.