On the robustness of dictatorships:

spectral methods.Ehud Friedgut,

Hebrew University, Jerusalem

Erdős-Ko-Rado (‘61)

• 407 links in Google

• 44 papers in MathSciNet with E.K.R. in the title (not including the original one, of course.)

The Erdős-Ko-Rado theorem

A fundamental theorem of extremal set theory:

Extremal example: flower.

Product-measure analogue

Extremal example: dictatorship.

The Ahlswede-Khachatriantheorem (special case)

Etc...

Or...

Or...

Product-measure analogue

Extremal example: duumvirate.

Beyond p < 1/3.

First observed and provenby Dinur and Safra.

From the measure-case to extremal set theory and back

Dinur and Safra proved the measure-results via

E.K.R. and Ahlswede-Khachatrian.

Here we attempt to prove measure-results using spectral methods, and deduce some corollaries in extremal set theory.

RobustnessA major incentive to use spectral

analysis on the discrete cube as a tool for

proving theorems in extremal set theory:

Proving robustness statements.

“Close to maximal size close to optimal structure”.

*

*Look for the purple star…

Intersection theorems,spectral methods…

Some people who did related work(there must be many others too):

Alon, Calderbank, Delsarte, Dinur,Frankl, Friedgut, Furedi, Hoffman,Lovász, Schrijver, Sudakov,Wilson...

Theorem 1

*

Corollary 1*

Theorem 2

*

Corollary 2*

t-intersecting familiesfor t>1

We will use the case t=2 to represent all t>1, the differences are merely

technical.

Digression:

Inspiration from a proof of a graph theoretic result

Spectral methods:Hoffman’s theorem

Hoffman’s theorem,sketch of proof

Sketch of proof, continued

Sketch of proof, continued

Sketch of proof, concluded

Stability observation:

Equality holds in Hoffman’s theorem only if the characteristic function of a maximal independent set is always a linear combination of the trivial eigenvector (1,1,...,1) and the eigenvectors corresponding to the minimal eigenvalue.

Also, “almost equality” implies “almost”the above statement.

Intersecting families and independent sets

Consider the graph whose vertices arethe subsets of {1,2,...,n}, with an edge between two vertices iff the correspondingsets are disjoint.

Intersecting family Independent set

Can we mimic Hoffman’s proof?

Problems...

• The graph isn’t regular, (1,1,...,1) isn’t an eigenvector.

• Coming to think of it, what are the eigenvectors? How can we compute them?

• Even if we could find them, they’re orthogonal with respect to the uniform measure, but we’re interested in a different product measure.

Let’s look at the adjacency matrix

Ø

Ø

{1}

{1}

Ø {1} {2} {1,2}

Ø{1}{2}{1,2}

This is good, because we can now computethe eigenvectors and eigenvalues of

But...

These are not the eigenvectors we want...

...However, looking back at Hoffman’sproof we notice that...

holds only because of the 0’s for non-edgesin A, not because of the 1’s. So...

Pseudo adjacency matrix

ReplaceØ

Ø

{1}

{1}

By

It turns out that a judicious choice is

Now everything works...

Their tensor products form an orthonormalbasis for the product space with the product measure, and Hoffman’s proof goes through (mutatis mutandis), yielding that if I is an independent set then μ(I)≤p.

Remarks...

It is associated with eigenvectorsof the type henceforth “first level eigenvectors”

This is the minimal eigenvalue,provided that p < ½ (!)

Boolean functions; Some facts of life

• Trivial : If all the Fourier coefficients are on levels 0 and 1 then the function is a dictatorship.

• Non trivial (FKN): If almost all the weight of the Fourier coefficients is on levels 0 and 1 then the function is close to a dictatorship.

• Deep (Bourgain, Kindler-Safra): Something similar is true if almost all the weight is on levels 0,1,…,k.

Remarks, continued...

• These facts of life, together with the “stability observation” following Hoffman’s proof imply the uniqueness and robustness of the extremal examples, the dictatorships .

• The proof only works for p< ½ ! (At p=1/2 the minimal eigenvalue shifts from one set of eigenvectors to another)

2-intersecting families

Can we repeat this proof for 2-intersecting

families?

Let’s start by taking a look at the adjacency

matrix...

The 2-intersecting adjacencymatrix

This doesn’tlook like the tensor productof smallermatrices...

Understanding the intersection matrices

The “0” in

(the 1-intersection matrix) warned us that when we add the same element to two disjoint sets they become intersecting.

Now we want to be more tolerant:

Different tactics for 2-intersecting

One common element= “warning”

But “two strikes, and yer out!’”

We need an element such that

Obvious solution:

Working over a ring

The solution: work over

Ø {1} {2} {1,2}Ø{1}{2}{1,2}

Ø {1}Ø

{1}

Now becomes...

2-Intersection matrix over

Working over a ring, continued...

• Same as before: we wish to replace

by some matrix to obtain the

“proper” eigenvectors.

• Different than before: the eigenvalues are now ring elements, so there’s no “minimal eigenvalue”.

Working over the ring, cont’d

Identities such as

Now become ,so, comparing coefficients, we canget a separate equation for the ηsand for the ρs…

…and after replacing the equalitiesby inequalities solve a L.P. problem

…More problems

However, the ηs and the ρs do not tensorseparately (they’re not products of the

coefficients in the case n=1 ).

Lord of the rings, part IIIIt turns out that now one has to know thevalue of n in advance before plugging thevalues into

If you plug in

a ***miracle*** happens...

2-intersecting - conclusion

...The solution of the L.P. is such that all the non-zero coefficients must belong only to thefirst level eigenvectors, or the second level eigenvectors.

Using some additional analysis of Boolean functions (involving [Kindler-Safra]) one may

finally prove the uniqueness and robustness result about duumvirates. Oh..., and the miracle breaks down at

p =1/3…

Questions...

• What about 3-intersecting families? (slight optimism.) • What about p > 1/3 ? (slight pessimism.)• What about families with no (heavy pessimism.)

• Stability results in coding theory and association schemes?...

?

Time will tell...

Have we struck a small gold mine...

...or just found a shiny coin?