Testing forbidden order-pattern properties on hypergrids

We study testing $\pi$-freeness of functions $f:[n]^d\to\mathbb{R}$, where $f$ is $\pi$-free if there there are no $k$ indices $x_1\prec\cdots\prec x_k\in [n]^d$ such that $f(x_i)<f(x_j)$ and $\pi(i) < \pi(j)$ for all $i,j \in [k]$, where $\prec$ is the natural partial order over $[n]^d$. Given $\epsilon\in(0,1)$, $\epsilon$-testing $\pi$-freeness asks to distinguish $\pi$-free functions from those which are $\epsilon$-far -- meaning at least $\epsilon n^d$ function values must be modified to make it $\pi$-free. While $k=2$ coincides with monotonicity testing, far less is known for $k>2$. We initiate a systematic study of pattern freeness on higher-dimensional grids. For $d=2$ and all permutations of size $k=3$, we design an adaptive one-sided tester with query complexity $O(n^{4/5+o(1)})$. We also prove general lower bounds for $k=3$: every nonadaptive tester requires $\Omega(n)$ queries, and every adaptive tester requires $\Omega(\sqrt{n})$ queries, yielding the first super-logarithmic lower bounds for $\pi$-freeness. For the monotone patterns $\pi=(1,2,3)$ and $(3,2,1)$, we present a nonadaptive tester with polylogarithmic query complexity, giving an exponential separation between monotone and nonmonotone patterns (unlike the one-dimensional case). A key ingredient in our $\pi$-freeness testers is new erasure-resilient ($\delta$-ER) $\epsilon$-testers for monotonicity over $[n]^d$ with query complexity $O(\log^{O(d)}n/(\epsilon(1-\delta)))$, where $0<\delta<1$ is an upper bound on the fraction of erasures. Prior ER testers worked only for $\delta=O(\epsilon/d)$. Our nonadaptive monotonicity tester is nearly optimal via a matching lower bound due to Pallavoor, Raskhodnikova, and Waingarten (Random Struct. Algorithms, 2022). Finally, we show that current techniques cannot yield sublinear-query testers for patterns of length $4$ even on two-dimensional hypergrids.