Ramsey games with giants

The classical result in the theory of random graphs, proved by Erdos and Renyi in 1960, concerns the threshold for the appearance of the giant component in the random graph process. We consider a variant of this problem, with a Ramsey flavor. Now, each random edge that arrives in the sequence of rounds must be colored with one of R colors. The goal can be either to create a giant component in every color class, or alternatively, to avoid it in every color. One can analyze the offline or online setting for this problem. In this paper, we consider all these variants and provide nontrivial upper and lower bounds; in certain cases (like online avoidance) the obtained bounds are asymptotically tight.Comment: 29 pages; minor revision

Coloring random graphs online without creating monochromatic subgraphs

Consider the following random process: The vertices of a binomial random graph $G_{n,p}$ are revealed one by one, and at each step only the edges induced by the already revealed vertices are visible. Our goal is to assign to each vertex one from a fixed number $r$ of available colors immediately and irrevocably without creating a monochromatic copy of some fixed graph $F$ in the process. Our first main result is that for any $F$ and $r$, the threshold function for this problem is given by $p_0(F,r,n)=n^{-1/m_1^*(F,r)}$, where $m_1^*(F,r)$ denotes the so-called \emph{online vertex-Ramsey density} of $F$ and $r$. This parameter is defined via a purely deterministic two-player game, in which the random process is replaced by an adversary that is subject to certain restrictions inherited from the random setting. Our second main result states that for any $F$ and $r$, the online vertex-Ramsey density $m_1^*(F,r)$ is a computable rational number. Our lower bound proof is algorithmic, i.e., we obtain polynomial-time online algorithms that succeed in coloring $G_{n,p}$ as desired with probability $1-o(1)$ for any $p(n) = o(n^{-1/m_1^*(F,r)})$.Comment: some minor addition

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 player's 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 follo

Fast construction on a restricted budget

We introduce a model of a controlled random graph process. In this model, the edges of the complete graph $K_n$ are ordered randomly and then revealed, one by one, to a player called Builder. He must decide, immediately and irrevocably, whether to purchase each observed edge. The observation time is bounded by parameter $t$, and the total budget of purchased edges is bounded by parameter $b$. Builder's goal is to devise a strategy that, with high probability, allows him to construct a graph of purchased edges possessing a target graph property $\mathcal{P}$, all within the limitations of observation time and total budget. We show the following: (a) Builder has a strategy to achieve minimum degree $k$ at the hitting time for this property by purchasing at most $c_kn$ edges for an explicit $c_k<k$; and a strategy to achieve it (slightly) after the threshold for minimum degree $k$ by purchasing at most $(1+\varepsilon)kn/2$ edges (which is optimal); (b) Builder has a strategy to create a Hamilton cycle if either $t\ge(1+\varepsilon)n\log{n}/2$ and $b\ge Cn$, or $t\ge Cn\log{n}$ and $b\ge(1+\varepsilon)n$, for some $C=C(\varepsilon)$; similar results hold for perfect matching; (c) Builder has a strategy to create a copy of a given $k$-vertex tree if $t\ge b\gg\{(n/t)^{k-2},1\}$, and this is optimal; and (d) For $\ell=2k+1$ or $\ell=2k+2$, Builder has a strategy to create a copy of a cycle of length $\ell$ if $b\gg\max\{n^{k+2}/t^{k+1},n/\sqrt{t}\}$, and this is optimal.Comment: 20 pages, 2 figure

Mini-Workshop: Positional Games

Positional games is one of rapidly developing subjects of modern combinatorics, researching two player perfect information games of combinatorial nature, ranging from recreational games like Tic-Tac-Toe to purely abstract games played on graphs and hypergraphs. Though defined usually in game theoretic terms, the subject has a distinct combinatorial flavor and boasts strong mutual connections with discrete probability, Ramsey theory and randomized algorithms. This mini-workshop was dedicated to summarizing the recent progress in the subject, to indicating possible directions of future developments, and to fostering collaboration between researchers working in various, sometimes apparently distinct directions

Hamiltonicity thresholds in Achlioptas processes

In this paper we analyze the appearance of a Hamilton cycle in the following random process. The process starts with an empty graph on n labeled vertices. At each round we are presented with K=K(n) edges, chosen uniformly at random from the missing ones, and are asked to add one of them to the current graph. The goal is to create a Hamilton cycle as soon as possible. We show that this problem has three regimes, depending on the value of K. For K=o(\log n), the threshold for Hamiltonicity is (1+o(1))n\log n /(2K), i.e., typically we can construct a Hamilton cycle K times faster that in the usual random graph process. When K=\omega(\log n) we can essentially waste almost no edges, and create a Hamilton cycle in n+o(n) rounds with high probability. Finally, in the intermediate regime where K=\Theta(\log n), the threshold has order n and we obtain upper and lower bounds that differ by a multiplicative factor of 3.Comment: 23 page