English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

We study Maker/Breaker games on the edges of the complete graph, asintroduced by Chvatal and Erdos. We show that in the (m:b) clique game playedon K_{N}, the complete graph on N vertices, Maker can achieve a K_{q} for q =(m/(log_{2}(b + 1)) - o(1)) * log N, which partially solves an open problem byBeck. Moreover, we show that in the (1:1) clique game played on K_{N} for asufficiently large N, Maker can achieve a K_{q} in only 2^(2q/3) moves, whichimproves the previous best bound and answers a question of Beck. Finally weconsider the so called tournament game. A tournament is a directed graph whereevery pair of vertices is connected by a single directed edge. The tournamentgame is played on K_{N}. At the beginning Breaker fixes an arbitrary tournamentT_{q} on q vertices. Maker and Breaker then alternately take turns at claimingone unclaimed edge e and selecting one of the two possible orientations. Makerwins if his graph contains a copy of the goal tournament T_{q}; otherwiseBreaker wins. We show that Maker wins the tournament game on K_{N} with q = (1- o(1))*log_{2}(N) which supports the random graph intuition: the threshold forq is asymptotically the same for the game played by two "clever'' players andthe game played by two ``random'' players.  This last result solves an open problem of Beck which he included in his listof the seven most humiliating open problems.

A Strategy for Maker in the Clique Game which Helps to Tackle some Open Problems by Beck