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 $x\in L$ can be written as $uvw$ such that $uv^iw\in L$ for all $i\geq 0$, then shouldn't there be some $i\geq 0$ such that every "sufficiently" long string $x\in L$ be of the form $uv^iw$ for some strings $u,v,w$?
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.