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The Clique Game. V í t Jel í nek, Jan K á ra, Robert Šámal Mentor: Dr. József Beck. The Rules of the Game. Two players take turns to color the edges of a complete graph on N vertices Each player wants to create a clique of size q with all its edges colored with his color
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The Clique Game Vít Jelínek, Jan Kára, Robert Šámal Mentor: Dr. József Beck
The Rules of the Game • Two players take turns to color the edges of a complete graph on N vertices • Each player wants to create a clique of size q with all its edges colored with his color • If all the edges are colored and no monochromatic clique of size q exists, the game is a draw
Observations • The second player can never have a winning strategy • If q is fixed and N islarge enough, draw is impossible (Ramsey theorem), so the first player has a winning strategy • Problems: • How to find an explicit winning strategy? • What if the board is smaller than the corresponding Ramsey number? • What if the board is infinite?
Our Project • We studied the K4 game on the infinite board • We found an explicit winning strategy for the first player • The proof is based on backtracking through the variation tree, improved by some observations about the game
Basic Idea of the Strategy • Step 1: make a triangle faster than your opponent (use our opening book) • Step 2: try to create a rabbit (see next slide) by adding edges connected to your triangle • Step 3: sooner or later, you reach a winning configuration (see next slide) and win
More Work to Do • Formulate the strategy and the proof in a clear and concise way • Try to further simplify the proof • Calculate the exact number of moves and vertices needed in the worst case • Write an article • Try to generalize to some Kn game, for n>4 (more ideas needed)
