1. Coloring graph powers; A Fourier approach N. Alon, I. Dinur, E. Friedgut, B. Sudakov

2. Traffic light Whenever you change all the switches......the light changes! How does that work?! Maybe... the light depends on only one switch?

3. Weak graph products

5. Coloring the product

8. Theorem: Trivial New Previously known (Lovász & Greenwell)

9. Extensions to general r -regular graphs This generalizes part (a)

10. Independent sets and the smallest eigenvalue

11. Theorem:

12. Sketch of the proof for the case of £ n K r For the sake of simplicity we will go through this proof for the case of r =3

13. Sketch of the proof for the case of £ n K r

17. easy

18. Sketch of the proof for the case of £ n K r Generalized F.K.N.

19. General r -regular graphs For the more general case we imitate this proof, and do pseudo-Fourier analysis on products of general graphs. Surprisingly enough, this amounts to no more than a change of basis in a linear space that allows us to “import” results such as F.K.N.

20. Highlights of the proof for the general case

22. From here on the proof proceeds almost precisely as before, we essentially “cut and paste” the previous arguments, where all the Fourier-related lemmas are preserved under the transformation between the two orthonormal bases of our space: the characters and the eigenvectors of G. (Crucially, this transformation has | S | $ | v |).

23. Questions?

24. Large independent sets Here is an example of a large independent set in f 0,1,2 g n : All vectors that have at least two 0’s among their first three coordinates. (The measure of this set is 7/27.) Are all reasonably large independent sets of similar form?

25. No, a random subset of such an independent set is also independent, yet does not depend on a fixed number of coordinates. However, we conjecture that the following is true:

26. Conjecture: Every large independent set is contained almost entirely in a junta. More Precisely:

27. Conjecture:

28. Part B: ( or The importance of being biased 1.1) Joint with Irit Dinur.

29. How to recover the junta? 0 12

30. The importance of being biased

31. The slope is equal to the sum of the influences

32. The junta lemma

33. Erdős-Ko-Rado (The sunflower theorem)

34. Corollary: Continuous asymptotic EKR.

35. From binary to ternary, the proof: Wait a minute, doesn’t that prove that every set is close to a junta according to some measure?!

36. Recovering the junta 0 12 0 12