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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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IV.19 Extremal and Probabilistic Combinatorics

Noga Alon and Michael Krivelevich

1 Combinatorics: An Introduction

1.1 Examples

It is hard to give a rigorous definition of combinatorics. Instead, let us start with a few examples to illustrate what the area is about.

(i) In the course of an examination of friendship between children some fifty years ago, the Hungarian sociologist Sandor Szalai observed that among any group of about twenty children he checked he could always find four children any two of whom were friends, or else four children no two of whom were friends. Despite the temptation to try to draw sociological conclusions, Szalai realized that this might well be a mathematical phenomenon rather than a sociological one. Indeed, a brief ...

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