By Dan Farkas
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Extra info for Linear Methods in Discrete Mathematics [Lecture notes]
Let p be a prime number and assume that n > 2p . Identify each subset of f1; 02; : : :1; ng containing p 0 1 elements with a vertex in the complete graph on p2n0 vertices. Color the edge joining two vertices labeled by subsets A and B red when #(A \ B ) 6 01 (mod p) : 2 2 1 38 Color all other edges blue. 1 In the complete graph on 0 n 1 p2 0 vertices with the coloring described above, there is no monochromatic complete subgraph on more than 0 n 1 2 1 p0 vertices. 1 1 Proof: Suppose there is a complete subgraph with red edges which uses m vertices.
1 4 1 n ! k n = k01 n0k+1 k ! s n n0s+1 k whenever k s. Set = n0ss . +1 4s n ) 3s n + 1 ) s n 0 2s + 1 ) n 0 2ss + 1 1 : In particular, < 1. Thus ! ! n n n 1 1 1 + + + s s01 0 ! (1 + + + 1 1 1 + ) ns ! n < (1 + + + 1 1 1) s ! = 1 01 ns ! n s = (1 + n 0 2s + 1 ) s ! n 2 s : 2 s 2 The rst application of this theorem will be to constructive Ramsey theory. To introduce the subject, we begin with a game and a question. Six dots are drawn on a piece of paper. Two players take turns drawing a line which connects two points, with the rules that no one can connect a pair of points already connected and one player uses a red crayon while the other uses blue.
We are done. ) 0 1 The bound we currently have can be rewritten n + (n + 1). We are 0 1 going to show that it can be signicantly improved to simply n . To establish this we will use fake dot products, genuine dot products, morethe-merrier, and calculus. Since we will be continuing the earlier argument, we borrow all of the notation. 1 Suppose that v; : : : ; v[k] are vectors in Rn such that the 2 1 2 ( 2 2 1 2 +1) 1 2 ( +1)( +4) 2 +2 2 +2 2 distance between any two of them is one of two positive numbers.
Linear Methods in Discrete Mathematics [Lecture notes] by Dan Farkas