Intersecting Families , Independent Sets and Coloring of Certain Graph Products

We study analytical methods applied to combinatorial problems. Speci cally, we use techniques from Fourier analysis in order to understand certain combinatorial structures. One of the main ideas of this approach is the following: Given a function of n variables, we wish to learn how a variable or a small subset of variables in uences the function, i.e. to understand how the change in a small number of variables a ects the value of the function. For some problems the goal is to nd a small number of variables, that the given function essentially depends only on them. In other problems we are interested in nding a variable that has a non-negligible in uence on the function. Part I The rst part of this thesis deals with a variant of the intersection theorem. Given r ≥ 3 we are interested in understanding the maximal possible size of F ⊆ [r]n, a family of strings of length n over the alphabet [r] = {0, 1 . . . r − 1}, so that every two strings in F agree in at least t coordinates, for some xed value of t. Alternatively one can think of this problem as the problem of characterizing independent sets of maximal size in the corresponding intersection graph, G = (V, E). The vertices of the graph are V = [r]n, and two vertices are disconnected if and only if the corresponding strings equal in at least t coordinates. We show that for certain range of r and t, the relative size of any t-intersecting family F is at most 1/rt. The maximum can be attained by taking all strings with some xed t coordinates. These extremal examples are not only unique but also stable: For any ≥ 0 any such F of size 1/r − can be approximated by some family de ned by xing exactly t coordinates. Part II In the second part we extend the result of Dinur et al. (STOC 2006) concerning the reduction from the 2-to-1 Label Cover problem to the Approximate Coloring problem. We show certain hardness of coloring a 4-colorable graph on n vertices with log(n) colors (for some constant c > 0). This is done by proving a variant of the Majority is Stablest Theorem of Mossel at el. (FOCS 2005), which gives a tight bound on the noise stability of boolean functions in which every coordinate has small in uence.

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