Synchronization Strings over the Optimal Alphabet

Abstract: Synchronization strings attach deterministic position labels to a stream that survive insertions and deletions. Haeupler and Shahrasbi introduced these objects, and subsequent work proved that four symbols suffice for some fixed parameter epsilon < 1, whereas two symbols cannot support arbitrarily long synchronization strings. We resolve the remaining ternary case: every length admits a ternary 2001/2002-synchronization string. Thus three is the exact minimum constant alphabet size. A computer-assisted refinement based on a larger 54-uniform family yields ternary epsilon-synchronization strings for every epsilon > 226/227. The previous four-symbol construction uses a ternary square-free backbone to exclude short repetitions and a fourth symbol to carry long-range synchronization marks. Our main technical contribution is a local-entropy transfer theorem: every square-free block-local source with positive conditional min-entropy supports synchronization strings with a fixed gap. We instantiate this theorem using occurrence-wise branching in a Brinkhuis family. Every outcome remains ternary and square-free, while every long interval retains linear conditional min-entropy after all choices outside it are exposed. A deletion-ball estimate converts this entropy into anti-concentration for high longest common subsequences, and an asymmetric Lovasz Local Lemma enforces all interval constraints simultaneously. The same framework also yields exponentially many valid words, a Las Vegas polynomial-time construction, synchronization circles, and synchronization within a class of extremal square-free words.