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 is large 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 K 4 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

Winning Configurations

Pruning the Variation Tree

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 K n game, for n>4 (more ideas needed)