{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,7,30]],"date-time":"2025-07-30T16:43:48Z","timestamp":1753893828200,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"2","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Waiter\u2013Client and Client\u2013Waiter games are two\u2013player, perfect information games, with no chance moves, played on a finite set (board) with special subsets known as the winning sets. Each round of the biased $(1:q)$ Waiter\u2013Client game begins with Waiter offering $q+1$ previously unclaimed elements of the board to Client, who claims one and leaves the remaining $q$ elements to be claimed by Waiter immediately afterwards. In a $(1:q)$ Client\u2013Waiter game, play occurs in the same way except in each round, Waiter offers $t$ elements for any $t$ in the range $1\\leqslant t\\leqslant q+1$. If Client fully claims a winning set by the time all board elements have been offered, he wins in the Client\u2013Waiter game and loses in the Waiter\u2013Client game. We give an estimate for the threshold bias (i.e. the unique value of $q$ at which the winner of a $(1:q)$ game changes) of the $(1:q)$ Waiter\u2013Client and Client\u2013Waiter versions of two different games: the non\u20132\u2013colourability game, played on the edge set of a complete $k$\u2013uniform hypergraph, and the $k$\u2013SAT game. More precisely, we show that the threshold bias for the Waiter\u2013Client and Client\u2013Waiter versions of the non\u20132\u2013colourability game is $\\frac{1}{n}\\binom{n}{k}2^{\\mathcal{O}_k(k)}$ and $\\frac{1}{n}\\binom{n}{k}2^{-k(1+o_k(1))}$ respectively. Additionally, we show that the threshold bias for the Waiter\u2013Client and Client\u2013Waiter versions of the $k$\u2013SAT game is $\\frac{1}{n}\\binom{n}{k}$ up to a factor that is exponential and polynomial in $k$ respectively. This shows that these games exhibit the probabilistic intuition.<\/jats:p>","DOI":"10.37236\/6290","type":"journal-article","created":{"date-parts":[[2020,1,10]],"date-time":"2020-01-10T15:54:08Z","timestamp":1578671648000},"source":"Crossref","is-referenced-by-count":1,"title":["Waiter\u2013Client and Client\u2013Waiter Colourability and $k$\u2013SAT Games"],"prefix":"10.37236","volume":"24","author":[{"given":"Wei En","family":"Tan","sequence":"first","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2017,6,30]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i2p46\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v24i2p46\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2020,1,17]],"date-time":"2020-01-17T04:56:02Z","timestamp":1579236962000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v24i2p46"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,6,30]]},"references-count":0,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2017,4,13]]}},"URL":"https:\/\/doi.org\/10.37236\/6290","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2017,6,30]]},"article-number":"P2.46"}}