The maths of connections University of St Andrews March 23rd, 2013
Connection 1: Friendship
The party problem Question How many people need to come to a party, to guarantee that at least three of them all know each other, or at least three of them are mutual strangers? Let s do an experiment! With this group of 6 people, we succeeded in finding a trio. What can we say, in general?
Introducing graphs We ll represent the people as dots, called vertices. If two people know each other, draw a red line (a red edge) between them. If two people don t know each other, draw a blue edge between them. Set of vertices and edges is a graph. We want to find the smallest number of vertices, such that however we colour the edges, we ll always find either a red or a blue triangle. If only 5 people are at our party, we can fail to find three friends or three strangers.
Six people suffice! The Pigeonhole Principle Suppose we have p pigeons and h holes, and let s be p/h (the squash factor ). If we put all of the pigeons in holes, then at least one hole contains at least s pigeons. Colva Pigeons other people Consider two holes: Knows Colva and Doesn t know Colva. At least one hole contains at least three pigeons.
I want more people to be friends! For 3 mutual friends or strangers we need 6 people. We write this R(3) = 6. What about four mutual friends or strangers? It s not too hard to show that R(4) = 18. The numbers R(n) are the Ramsey numbers: Ramsey proved that R(n) is finite for all n. Frank Ramsey What about R(5)? The best we can say is 43 R(5) 49. We know that 102 R(6) 165. Paul Erdős
An infinite party? Infinite Ramsey Theorem Given infinitely many vertices, we can always find infinitely many connected all with red edges or all with blue edges. A prime number is a number that is divisible only by itself and 1. Every positive number factorises uniquely into primes: 12 = 2 2 3. A Number-Theoretic Consequence There exists an infinite set S of positive whole numbers, such that for all pairs m, n of numbers in S, the sum m + n has an even number of prime factors, including multiplicity. Infinitely many red edges: all fine. Infinitely many blues: double! Challenge: Find such a set!
Connection 2: Marriage
A theorem on marriage Hall s Marriage Theorem (Phillip Hall, 1935) Consider n women, W 1,..., W n. Suppose each woman W i has a list M i of the men she would happily marry. (Each man will happily marry any woman who wants him.) Every woman can be happily married if and only if for each set W of women, the union of their lists M i of men contains at least W men.
An application: Sudoku The Independent: There s no mathematics involved. Use logic and reasoning to solve the puzzle. 2 1 9 2 4 8 2 9 1 2 9 5 3 3 4 8 3 1 2 6 1 5 2 3 5 8 7 Consider the bottom left sub-square. The women are the unfilled cells: (7, 1), (7, 2), (7, 3), (8, 1) and (9, 3). The men are the unused numbers: 4, 6, 7, 8, 9.
An application: Sudoku The Independent: There s no mathematics involved. Use logic and reasoning to solve the puzzle. 2 1 9 2 4 8 2 9 1 2 9 5 3 3 4 8 3 1 2 6 8 1 5 2 3 7 5 8 7 (7, 1) (7, 2) (7, 3) (8, 1) (9, 3) 4 6 7 8 9 A critical set W of women, is a set that are willing to marry exactly W men. We find critical sets of women, and use them.
Longer distance connections
The Kevin Bacon Game Given an actor, find the shortest path from them to Kevin Bacon, using only films in the Internet Movie Database (IMDB). The minimum number of films is the actor s Bacon number. was in The big picture with John Cleese Kevin Bacon Let s make a graph! Vertices: every actor in IMDB. Edges: between actors who ve appeared in a film together. Bacon number is the number of edges in a path from actor to Bacon. (Infinity if no path exists.)
Bacon and more Candy Brando 1968 Skum Rocks Ringo 2013 Bacon The Erdo s graph has as vertices everyone who s published an academic paper. Two people have an edge between them if they have published together. Nina Colva Max Akos Seress Erdo s
Erdős Bacon numbers? Erdős number + Bacon number = Erdős Bacon number. Most people s Erdős Bacon number is infinity. Someone with an Erdős Bacon number of 6 is: Natalie Portman
Connection 3: Disease
The spread of disease Imagine you re a farmer, planting an apple orchard. Blight can arrive in your orchard, via birds and insects. Once there, it can spread on the wind from a tree to its neighbours. More trees means more apples, but the closer together the trees are planted, the more likely it is that blight will spread. Question How closely should you plant the trees?
Some experiments Let s run some experiments on a big square grid. p = 1/4 p = 1/2 p = 3/4 With p = 1/4, one infection spreads to at most 4 other trees. With p = 1/2, the biggest cluster has size 36, and more than half the trees are in three big clusters. With p = 3/4, one infection covers almost the whole orchard.
Percolation theory The study of problems like this is percolation theory. Instead of edges, we can percolate through the vertices: make them open with probability p and closed otherwise. Applications of percolation theory include: The spread of wild fires. The spread of human diseases: flu and SARS.
Infinity and probability For many applications we consider infinite sets of vertices. The system percolates if after putting in the edges with fixed probability p, infinitely many vertices are all joined together. Infinite orchard: You might think that the probability of an infinite set of infected trees changes smoothly from 0 to 1, as we plant the trees closer together. In fact, the probability of an infinite set of infected trees jumps like this. The point where it jumps is the critical probability, p c.
The critical probability In 1960, Ted Harris proved that pc 1/2 for our infinite grid. In 1980, Harry Kesten proved that pc is exactly 1/2. We d love to know more about what happens when p is exactly pc. Wendelin Werner and Stanislav Smirnov both won Fields medals for work relating to percolation theory.
Why do the galaxies look like they do? One answer (Schulman & Seiden): Each bit of a galaxy contains stars, and regions of gas, that could collapse to form stars. Something needs to happen to trigger that collapse. A supernova! This sends a shockwave through space, triggering star formation. Many years later, some of these new stars will go nova in their turn: galactic percolation.