The Friendship Theorem Dr. John S. Caughman Portland State University

1. The Friendship TheoremDr. John S. CaughmanPortland State University

2. Public Service Announcement “Freshman’s Dream” (a+b)p=ap+bp …mod p …when a, b are integers …and p is prime.

3. Freshman’s Dream Generalizes! (a1+a2+…+an)p=a1p +a2p +…+anp …mod p …when a, b are integers …and p is prime.

4. a1 * * * 0 a2 * * A = 0 0 a3 * 0 0 0 a4 Freshman’s Dream Generalizes (a1+a2+…+an)p = a1p +a2p +…+anp tr(A) p = tr(Ap) (mod p)

5. * * * * * * * * A = * * * * * * * * Freshman’s Dream Generalizes! tr(A) p = tr(Ap) (mod p) tr(A p)= tr((L+U)p) =tr(Lp +Up) = tr(Lp)+tr(Up)=0+tr(U)p = tr(A)p Note:tr(UL)=tr(LU) so cross terms combine , and coefficients =0 mod p.

6. The Theorem If every pair of people at a party has precisely one common friend, then there must be a person who is everybody's friend.

7. Cheap Example Nancy John Mark

8. Cheap Example of a Graph Nancy John Mark

9. What a Graph IS: Nancy John Mark

10. What a Graph IS: Nancy Vertices! John Mark

11. What a Graph IS: Nancy Edges! John Mark

12. What a Graph IS NOT: Nancy John Mark

13. What a Graph IS NOT: Loops! Nancy John Mark

14. What a Graph IS NOT: Loops! Nancy John Mark

15. What a Graph IS NOT: Nancy Directed edges! Mark John

16. What a Graph IS NOT: Nancy Directed edges! Mark John

17. What a Graph IS NOT: Nancy Multi-edges! John Mark

18. What a Graph IS NOT: Nancy Multi-edges! John Mark

19. ‘Simple’ Graphs… Nancy • Finite • Undirected • No Loops • No Multiple Edges John Mark

20. The Theorem, Restated Let G be a simple graph with n vertices. If every pair of vertices in G has precisely one common neighbor, then G has a vertex with n-1 neighbors.

21. The Theorem, Restated Generally attributed to Erdős (1966). Easily proved using linear algebra. Combinatorial proofs more elusive.

22. NOT A TYPICAL “THRESHOLD” RESULT

23. Pigeonhole Principle If more than n pigeons are placed into n or fewer holes, then at least one hole will contain more than one pigeon.

24. Some threshold results If a graph with n vertices has > n2/4 edges, then there must be a set of 3 mutual neighbors. If it has > n(n-2)/2 edges, then there must be a vertex with n-1 neighbors.

25. Extremal Graph Theory If this were an extremal problem, we would expect graphs with MORE edges than ours to also satisfy the same conclusion…

30. 1

31. 1 2

32. 1 2 3

33. 1 4 2 3

34. 4

35. 4

36. 4

37. 2

38. Of the 15 pairs, 3 have four neighbors in common and 12 have two in common. So ALL pairs have at least one in common. But NO vertex has five neighbors!

39. Related Fact – losing edges

40. Related Fact – losing edges

41. Related Fact – losing edges

42. Summary If every pair of vertices in a graph has at least one neighbor in common, it mightnot be possible to remove edges and produce a subgraph in which every pair has exactly one common neighbor.

43. Accolades for Friendship • The Friendship Theorem is listed among Abad's “100 Greatest Theorems” • The proof is immortalized in Aigner and Ziegler's Proofs from THE BOOK.

44. Example 1

45. Example 2

46. Example 3

47. How to prove it: STEP ONE: If x and y are not neighbors, they have the same # of neighbors. Why: Let Nx = set of neighbors of x Let Ny = set of neighbors of y

48. y x How to prove it:

49. y x Nx How to prove it:

50. y x Ny How to prove it: