Assignment2.pdf - Introduction to Combinatorics Assignment 2 University of Toronto Scarborough Name Student ID \u2022 Please submit your homework on

# Assignment2.pdf - Introduction to Combinatorics Assignment...

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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. 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
Problem 3. (10 points) Prove, using the principle of extremals, that the number 24 is irrational. Explain why your argument does not show that 25 is irrational.

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