Upper Bounds for Online Ramsey Games in Random Graphs

Abstract

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 follow.

Offline thresholds for Ramsey‐type games on random graphs

“…(In fact, it was shown in [9] that for any ≥ 2 and any non-forest , 2−1/ 2 ( ) is indeed a lower bound for the threshold of the game with colors. However, the upper bound proof presented in [10] does not extend to the game with more than two colors. )…”

“…The motivation for this work comes from three Ramsey-type one-player games that have been studied in an online setting in previous work: the online Ramsey game [2,9,10], the balanced online Ramsey game [7,12], and the Achlioptas game [5,11]. In all three games, the edges of the complete graph appear in a random order, either one by one or in batches of some fixed size.…”

“…This game was introduced in [2] for the case = 3 and = 2 colors. In [9,10], the following theorem was shown.…”

“…Theorem 1 ( [9,10]). Let be a non-forest that has a subgraph − ⊂ with −1 edges satisfying This result applies in particular to cliques and cycles of arbitrary size.…”

Ramsey properties of random discrete structures

“…On the other hand, the study of Ramsey type properties of random structures was initiated only more recently by Luczak, Ruciński, and Voigt [19] and further studied by Rödl and Ruciński with their collaborators [7,9,10,26,27,28,29,30,31] (for more related results by others see [8,17,18,20,21,22,23]). The aim of this paper is to establish a general result which yields Ramsey type results for random discrete structures (see Theorem 2.5).…”