# automata – Implication of the Pumping lemma

I’m reading Hopcroft and Ullman’s ’79 edition of “Introduction to Automata theory, Languages, and Computation”. In chapter 3, the authors say “The lemma(sic) does not state that every sufficiently long string in a regular set is of the form $$uv^iw$$ for some large $$i$$“. I can’t see how this is true.

If I’m not wrong, the pumping lemma states that if $$L$$ is a regular set, then every “sufficiently” long string $$xin L$$ can be written as $$uvw$$ such that $$uv^iwin L$$ for all $$igeq 0$$, then shouldn’t every “sufficiently” long string $$xin L$$ be of the form $$uv^iw$$ for some strings $$u,v,w$$ and for some $$igeq 0$$?

Also, the authors state (whose proof is left as an exercise) that the set $$(0+1)^*$$ contains arbitrarily long strings in which no substring appears three times consecutively. I can’t seem to see how this is true as well.

Any help or perhaps even a hint is well appreciated.