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

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Presentation on theme: "Coloring graph powers; A Fourier approach N. Alon, I. Dinur, E. Friedgut, B. Sudakov."— Presentation transcript:

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

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5 Coloring the product

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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

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15

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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

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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

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