Waiter–Client and Client–Waiter games

In this thesis, we consider two types of positional games; $Waiter$-$Client$ and $Client$-$Waiter$ games. Each round in a biased ($a$:$b$) game begins with Waiter offering a+b free elements of the board to Client. Client claims $a$ elements among these and the remaining $b$ elements are claimed by Waiter. Waiter wins in a Waiter-Client game if he can force Client to fully claim a $winning$ $set$, otherwise Client wins. In a Client-Waiter game, Client wins if he can claim a winning set himself, else Waiter wins. We estimate the $threshold$ $bias$ of four different ($1$:$q$) Waiter-Client and Client-Waiter games. This is the unique value of Waiter's bias $q$ at which the player with a winning strategy changes. We find its asymptotic value for both versions of the complete-minor and non-planarity games and give bounds for both versions of the non-$r$-colourability and $k$-SAT games. Our results show that these games exhibit a heuristic called the $probabilistic$ $intuition$. We also find sharp probability thresholds for the appearance of a graph in the random graph $G$($n$,$p$) on which Waiter and Client win the ($1$:$q$) Waiter-Client and Client-Waiter Hamiltonicity games respectively