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Abstract for Aviezri S.\ Fraenkel and 
R.\ Jamie Simpson,
How Many Squares Must a Binary Sequence Contain?


Let $g(n)$ be the length of a longest binary
string containing at most $n$ distinct squares (two identical adjacent
substrings). Then $g(0)=3$ (010 is such a string), $g(1)=7$ (0001000) and
$g(2)=18$ (010011000111001101). How does the sequence $\bigl\{g(n)
\bigr\}$ behave? We give a complete answer.
 
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