0-1 String Problem

I recently had a problem in my mind and am having some trouble proving my solution, have a look.

Imagine I make a string of numbers with only 1's and 0's. ex: \(10101010000111010101...\)

\(Q:\) How many numbers (1's and 0's) would I have to write (at least) to guarantee a repetition of any \(n\)-string number. Ex: Let \(n=2\), generate a random sequence of 1's and 0's: \(100110\). Notice that the first 2 digits are "\(10\)", so is the 5th and 6th "\(10\)" a repetition!

For \(n=2\), I have proved a string of length \(>5\) must have at least one repetition. For \(n=2\) we have answer \(5\). Similarly, for \(n=3\), we found the answer to be \(10\), the string length cannot exceed \(10\) without repeating a \(3\)-string number. I couldn't find a number for \(n=4\) but I have shown that for any \(n\) the string length does not exceed \( 2^n+n-1\) but I suspect \( 2^n+n-1\) might be the general formula (If you substitute \(n=2\) and \(n=3\) you will find the results match), but I haven't been able to prove this for all \(n\).

PS: I think the solution might be related to graph theory.

Note by Apratim Ghosh
6 months, 2 weeks ago

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This is a well-studied problem. Maybe try OEIS first next time: http://oeis.org/A052944

Richard Xu - 5 months, 3 weeks ago

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For n=2, surely this is 6?

Stephen Mellor - 6 months, 2 weeks ago

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