Grasshopper Avoidance of Patterns

Motivated by a geometrical Thue-type problem, we introduce a new variant of the classical pattern avoidance in words, where jumping over a letter in the pattern occurrence is allowed. We say that pattern $p\in E^+$ occurs with jumps in a word $w=a_1a_2\ldots a_k \in A^+$, if there exist a non-erasing morphism $f$ from $E^*$ to $A^*$ and a sequence $(i_1, i_2, \ldots , i_l)$ satisfying $i_{j+1}\in\{ i_j+1, i_j+2 \}$ for $j=1, 2, \ldots, l-1$, such that $f(p) = a_{i_1}a_{i_2}\ldots a_{i_l}.$ For example, a pattern $xx$ occurs with jumps in a word $abdcadbc$ (for $x \mapsto abc$). A pattern $p$ is grasshopper $k$-avoidable if there exists an alphabet $A$ of $k$ elements, such that there exist arbitrarily long words over $A$ in which $p$ does not occur with jumps. The minimal such $k$ is the grasshopper avoidability index of $p$. It appears that this notion is related to two other problems: pattern avoidance on graphs and pattern-free colorings of the Euclidean plane. In particular, we show that a sequence avoiding a pattern $p$ with jumps can be a tool to construct a line $p$-free coloring of $\mathbb{R}^2$. In our work, we determine the grasshopper avoidability index of patterns $\alpha^n$ for all $n$ except $n=5$. We also show that every doubled pattern is grasshopper $(2^7+1)$-avoidable, every pattern on $k$ variables of length at least $2^k$ is grasshopper $37$-avoidable, and there exists a constant $c$ such that every pattern of length at least $c$ on $2$ variables is grasshopper $3$-avoidable (those results are proved using the entropy compression method).