A set S of six points P1, P2, ..., P6 in a plane that can form the vertices of a convex hexagon is given. All possible pairs of points from the set S are randomly selected and each time joined together randomly either by a blue string or a red string. Find the probability that there will be a triangle (with vertices from the set S) whose string sides have all the same color.

prob of 1.really enjoyed reading about Ramsey theory. here is a game based on it:same setup, but now two players alternatively color these strings by either blue or red. whoever completes a triangle whose sides have the same color wins. because the procedure ends up with a same-color triangle for sure, there would be no tie. so in order to win, you prefer to go first or second?

starter needs max 4 steps to get a triangle and it works for 5 points as well

arkol, could you elaborate your statement a little more? thx.

first to go wins.assuming that either party doesnt make unforced errors, i.e. if for e.g. verticies are 1 to 6, say 1-2 and 1-3 are colored blue then red is forced to move 2-3. if red doesnt move 2-3 then lets call it unforced error.either player is trying to check mate the other. the easiest check mate is when one party can complete (atleast) two triangles in the next move.say blue starts.Blue: 1-2Red: choose any possible moveBlue: choose any possible of {1-3,1-4,1-5,1-6}Red: forced move and this move necessarily doesnt belong to {1-2,1-3,1-4,1-5,1-6}Blue: choose any possible of {1-3,1-4,1-5,1-6}Red is check mated.this is generalizable, since the result is independent of the starting move, sort of symetric across the possible first moves.

both players can color the strings by either blue or red. there is no restriction that one player has to color red, and the other has to color blue.----- ----- ----- ----- -----if one player only colors in red and the other colors in blue, and whoever completes a same-color triangle loses, then it is the so-called "Sim" game.