Analysis of Boolean Functions – week 1

Home page of the course.

In the first lecture I defined the discrete n-dimensional cube and  Boolean functions. Then I moved to discuss five problems in extremal combinatorics dealing with intersecting families of sets.

1) The largest possible intersecting family of subsets of [n];

2) The largest possible intersecting family of subsets of [n] so that the family of complements is also intersecting;

3) The largest possible family of graphs on v vertices such that each two graphs in the family contains a common triangle;

4) Chvatal’s conjecture regarding the maximum size of an intersecting family of sets contained in an ideal of sets;

5) Erdos-Ko-Rado Theorem.

Exercise: Prove one of the following

a) The Harris-Kleitman’s inequality

b) (from the H-K inequality) Every family of subsets of [n] with the property that every two sets have non-empty intersection and no full union contains at most 2^{n-2} sets.

More reading: this post :”Extremal combinatorics I: extremal problems on set systems“. Spoiler: The formulation of Chvatal’s conjecture but also the answer to the second exercise can be found on this post: Extremal combinatorics III: some basic theorems. (See also peoblen 25 in the 1972 paper Selected combinatorial research problems by Chvatal, Klarner and Knuth.)


I moved to discuss the problem of collective coin flipping and the notion of influence as defined by Ben-Or and Linial. I mentioned the Baton-passing protocol, the Alon-Naor result, and Feige’s two-rooms protocol.

More reading: this post :” Nati’s influence“. The original paper of Ben-Or Linial:  Collective coin flipping, M. Ben-Or  and N. Linial, in “Randomness and    Computation” (S. Micali ed.) Academic Press, New York, 1989, pp.    91-115.

Nati’s Influence

When do we say that one event causes another? Causality is a topic of great interest in statistics, physics, philosophy, law, economics, and many other places. Now, if causality is not complicated enough, we can ask what is the influence one event has on another one.  Michael Ben-Or and Nati Linial wrote a paper in 1985 where they studied the notion of influence in the context of collective coin flipping. The title of the post refers also to Nati’s influence on my work since he got me and Jeff Kahn interested in a conjecture from this paper.


The word “influence” (dating back, according to Merriam-Webster dictionary, to the 14th century) is close to the word “fluid”.  The original definition of influence is: “an ethereal fluid held to flow from the stars and to affect the actions of humans.” The modern meaning (according to Wictionary) is: “The power to affect, control or manipulate something or someone.”

Ben-Or and Linial’s definition of influence

Collective coin flipping refers to a situation where n processors or agents wish to agree on a common random bit. Ben-Or and Linial considered very general protocols to reach a single random bit, and also studied the simple case where the collective random bit is described by a Boolean function f(x_1,x_2,\dots,x_n) of n bits, one contributed by every agent. If all agents act appropriately the collective bit will be ‘1’ with probability 1/2. The purpose of collective coin flipping is to create a random bit R which is immune as much as possible against attempts of one or more agents to bias it towards ‘1’ or ‘0’. Continue reading