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 ∈ E+ occurs with jumps in a word w = a1a2 . . . ak ∈ A+, if there exist a non-erasing morphism f from E∗ to A∗ and a sequence (i1, i2, . . . , il) satisfying ij+1 ∈ {ij + 1, ij + 2} for j = 1, 2, . . . , l − 1, such that f(p) = ai1ai2 . . . ail . For example, a pattern xx occurs with jumps in a word abdcadbc (for x 7→ 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 R2. In our work, we determine the grasshopper avoidability index of patterns αn for all n except n = 5. We also show that every doubled pattern is grasshopper (27 + 1)-avoidable, every pattern on k variables of length at least 2k 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).