Reto Spöhel

Online ramsey games in random graphs

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The players goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove a lower bound of nβ(F,r) on the typical duration of this game, where β(F,r) is a function that is strictly increasing in r and satisfies limr→∞ β(F,r) = 2 − 1/m2(F), where n2−1/m2(F) is the threshold of the corresponding offline colouring problem.

Upper bounds for online ramsey games in random graphs

Consider the following one-player game. Starting with the empty graph on n vertices, in every step a new edge is drawn uniformly at random and inserted into the current graph. This edge has to be coloured immediately with one of r available colours. The players goal is to avoid creating a monochromatic copy of some fixed graph F for as long as possible. We prove an upper bound on the typical duration of this game if F is from a large class of graphs including cliques and cycles of arbitrary size. Together with lower bounds published elsewhere, explicit threshold functions follow.

Playing Mastermind With Many Colors

We analyze the general version of the classic guessing game Mastermind with n positions and k colors. Since the case k ≤ n1 − ε, ε > 0 a constant, is well understood, we concentrate on larger numbers of colors. For the most prominent case k = n, our results imply that Codebreaker can find the secret code with O(nlog log n) guesses. This bound is valid also when only black answer pegs are used. It improves the O(nlog n) bound first proven by Chvátal. We also show that if both black and white answer pegs are used, then the O(nlog log n) bound holds for up to n2log log n colors. These bounds are almost tight, as the known lower bound of Ω(n) shows. Unlike for k ≤ n1 − ε, simply guessing at random until the secret code is determined is not sufficient. In fact, we show that an optimal nonadaptive strategy (deterministic or randomized) needs Θ(nlog n) guesses.

Probabilistic One-Player Ramsey Games via Deterministic Two-Player Games

Consider the following probabilistic one-player game: The board is a graph with

PAC learning and genetic programming

n

Upper bounds on probability thresholds for asymmetric Ramsey properties

vertices, which initially contains no edges. In each step, a new edge is drawn uniformly at random from all nonedges and is presented to the player, henceforth called Painter. Painter must assign one of

Playing mastermind with many colors

r

Online vertex-coloring games in random graphs

available colors to each edge immediately, where

The online clique avoidance game on random graphs

r\geq 2

Coloring random graphs online without creating monochromatic subgraphs

is a fixed integer. The game is over as soon as a monochromatic copy of some fixed graph