Unavoidable Sets

Recently, Blanchet-Sadri, Flapan, and Watkins investigated the minimum number of holes in unavoidable sets of uniform length. They conjectured that the \(m\) uniform set \(X= \{a{\diamond}^{m-2}a,a{\diamond}^{m-2}c,b{\diamond}^{m-2}b, c{\diamond}^{m-2}c, a{\diamond}^{x_1}b{\diamond}^{x_2}b, b{\diamond}^{y_1}b{\diamond}^{y_2}c \}\) is avoidable for all \(m,x_1,x_2,y_1,y_2\). Given that the conjecture is true, Blanchet-Sadri et al. provided an exact formula for the minimum number of holes in uniform unavoidable sets. We attempt to prove the conjecture by taking a constructive approach. We provide specific infinite words that avoid this set under certain conditions on \(m,x_1,x_2,y_1,y_2\).

Keywords: Combinatorics on words, partial words, unavoidables sets.