Algebraic Combinatorics I: Association Schemes (Mathematics by Eiichi Bannai

By Eiichi Bannai

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Extra resources for Algebraic Combinatorics I: Association Schemes (Mathematics lecture note series) (Bk. 1)

Example text

Let s be the number of shapes used of the first two types. Find the largest possible value of s. 19 We are given a 5 × 5 board with a chessboard coloring where the corners are black. In each black square there is a black token and in each white square there is a white token. The tokens can move to neighbor squares (squares that share a side with the one they are on). A and B are going to play by turns in the following way: First A chooses a black token and removes it from the board. Then, A moves a white token to the empty space.

10 (OMM 1998) The sides and diagonals of a regular octagon are colored black or red. Show that there are at least 7 monochromatic triangles with vertices in the vertices of the octagon. 11 (IMO 1964) 17 people communicate by mail with each other. In all their letters they only discuss one of three possible topics. Each pair of persons discusses only one topic. Show that there are at least three persons that discussed only one topic. 12 Show that if l, s are positive integers, then r(l, s) ≤ l+s −2 .

This means that A and B divide the circle in two arcs, one of which has an odd number of persons and the other an even number. The strategy for A is to remove always a person on the even side. This leaves B with an odd number of persons on each side. When B plays, A has again a side with an odd number of persons and an even number on the other, so he can continue with his strategy. B can never hope to win if A plays this way, since he always has at least one person between himself and A on both sides.

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