Tarski Lower Bounds from Multi-Dimensional Herringbones

Tarski's theorem states that every monotone function from a complete lattice to itself has a fixed point. We analyze the query complexity of finding such a fixed point on the $k$-dimensional grid of side length $n$ under the $\leq$ relation. In this setting, there is an unknown monotone function $f: \{0,1,\ldots, n-1\}^k \to \{0,1,\ldots, n-1\}^k$ and an algorithm must query a vertex $v$ to learn $f(v)$. The goal is to find a fixed point of $f$ using as few oracle queries as possible. We show that the randomized query complexity of this problem is $\Omega\left( \frac{k \cdot \log^2{n}}{\log{k}} \right)$ for all $n,k \geq 2$. This unifies and improves upon two prior results: a lower bound of $\Omega(\log^2{n})$ from [EPRY 2019] and a lower bound of $\Omega\left( \frac{k \cdot \log{n}}{\log{k}}\right)$ from [BPR 2024], respectively.