Assignment2_solutions.pdf - Introduction to Combinatorics Assignment 2 University of Toronto Scarborough Name Student ID \u2022 Please submit your homework

# Assignment2_solutions.pdf - Introduction to Combinatorics...

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Introduction to Combinatorics: Assignment 2 University of Toronto Scarborough Name: Student ID: Please submit your homework on gradescope online. Write your solution to all seven problems. Due date: February 16, 6pm 1
Problem 1. (5 points) In a chess tournament there are n players participating. Every player has to play. Initially, every player has 0 points. Winners get 3 points, losers get -1 point (lose 1 point) and players who tie get 1 point each. If every player has to play with every other player, what is the sum of points of all players at the end of the tournament? Solution. After every game 2 points are added. There are n ( n +1) / 2 games. Hence n ( n + 1) points. 2
Problem 2. (15 points) A. Define the Ramsey numbers R ( m, n ) for m, n N . B. Show that if the edges of a complete graph with 17 vertices are colored either red, or green or blue then there must be a monochromatic triangle. Solution. 3
A. Lecture notes B. Consider one of the vertices and its adjacent 16=3 · 5+1 edges. By PP there are 6 edges with the same color, say green. If one of the other edges (the ones connecting the

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