Nivat's conjecture

Let $D\subseteq\mathbb{Z}^2$ be a rectangle and $c\colon\mathbb{Z}^2\to\mathcal A$ a configuration with $P_c(D)\leq\lvert D\rvert$.

Nivat's conjecture. Then $c$ is $1$-periodic.

This is the two-dimensional analogue of the Morse--Hedlund theorem and predicts that sufficiently low local pattern complexity forces global periodicity. The conjecture remains open, but strong partial results are known.