Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, May 26, 2015, 12:15 pm

Duration: 45 minutes

Location: OAT S15/S16/S17

Speaker: Raúl Penaguião

Non-repetitive sequences and games in paths and graphs

A non-repetitive sequence is a sequence of symbols in which no two consecutive blocks of the same size are equal. 123132123 is nonrepetitive. With two symbols it's impossible to write an infinite non-repetitive sequence, but a remarkable result due to Thue asserts that such is possible using only three distinct symbols. This paper introduces a more general problem, where one is only allowed to use a symbol from a previously given list L_i on the position i. It's always possible to construct in such settings a non-repetitive infinite sequence when the given lists are of size four, and it's still a conjecture weather for lists of size three still holds. This is proven using a randomized algorithm. In this context, a games is introduced. The erase-repetition game is a game played by two people where a sequence is constructed alternately according to the symbols in the lists L_i, and player A tries to build an arbitrary long nonrepetitive sequence, whereas player B tries to keep the sequence on a small size. We are able to prove that the player A has a strategy, provided that the lists are big enough.

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