parikshit gopalan georgia institute of technology atlanta, georgia, usa

Constructing Ramsey GraphsConstructing Ramsey Graphsfromfrom

Boolean Function Boolean Function RepresentationsRepresentations..

Parikshit GopalanParikshit Gopalan

Georgia Institute of TechnologyGeorgia Institute of Technology

Atlanta, Georgia, USA.Atlanta, Georgia, USA.

Explicit Ramsey Graph Constructions

[Erdös] : There exists a graph G on 2n vertices with (G), (G) · 2n.

Probabilistic Method.$100 for explicit construction.

[Ramsey] : Every graph on 2n vertices has either an independent set or a clique of size n/2.

Easy to construct G on 2n vertices with (G), (G) · 2n/2.

Alternate ViewConstructing Ramsey graphs:

Two color the edges of Kn so that there are no large monochromatic cliques.

Constructing Multicolor Ramsey graphs:

Color the edges of Kn using t colors so that there are no large monochromatic cliques.

A Brief History of Explicit Constructions

[Frankl-Wilson] : Gives (G), (G) · 2√n.• Extremal set theory.

[Grolmusz] : Same bound, multicolor graphs.• Polynomial representations of the OR function.

[Alon] : Similar to Frankl-Wilson, multicolor graphs.• Polynomial representations of graphs.

[Barak-Rao-Shaltiel-Wigderson] : (G), (G) · 2n.

• Extractors and pseudorandomness.

Polynomial Representations of Boolean functions

Def: P(X1,…,Xn) over Zm represents f: {0,1}n ! {0,1} if

f(x) f(y) ) P(x) P(y) mod m

Lower bounds for AC0[m].

Prime Case: [Razborov, Smolensky] :• Small circuits ≈ Low-degree polynomials.• Prove degree lower bounds.

Composite Case:Low-degree polynomials ) Small circuitsDegree lower bounds over Zm. (Simpler problem?)

Representing the OR function

Problem: What is the degree of OR mod m ?

For p prime: (n).

For m composite (say 6):

• Conjecture: (n) [Barrington]

• O(n1/2) upper bound. [Barrington-Beigel-Rudich]

• (log n) lower bound. [Barrington-Tardos]

Can asymmetry help compute a symmetric function?

[Barrington-Beigel-Rudich, Grolmusz, Tsai, Barrington-Tardos, Green, Alon-Beigel, Bhatnagar-G.-Lipton, Hansen]

A Connection [Grolmusz]

Problem: Let F be a family of subsets Si of [n] where

|Si| = 0 mod m

|Si Å Sj| 0 mod m

How large can F be?

Grolumsz: If m = 6, |F| can be superpolynomial in n.

Uses O(√n) degree OR polynomial of BBR.

Gives a Ramsey graph matching FW.

Better OR polynomials ) Better graphs.

Our Results

New view of OR representations.

Simple Ramsey construction from OR representations.

Unifies Frankl-Wilson, Grolmusz, Alon. All based on O(√n) symmetric OR polynomials.

Consequences : Insight from complexity: Asymmetry versus Symmetry Extends to multicolor Ramsey graphs. Improved bounds for restricted set systems.

Outline of This Talk

I Ramsey graphs from OR representations.• New view of OR representations.• Sample constructions.• Ramsey graphs.

II Limitations to Symmetric Constructions.

Outline of This Talk

I Ramsey graphs from OR representations.• New view of OR representations.• Sample constructions.• Ramsey graphs.

II Limitations to Symmetric Constructions.

OR Representations

New view of an OR representation:

Two polynomials s.t. the union of their zero sets is {0,1}n \ {0}.

P = 0 Q = 0

OR Representations

New view of an OR representation:• Two polynomials. • Union of their zero sets is {0,1}n \ {0}. • Degree of representation = max(deg(P), deg(Q)).

Both polynomials mod p. P mod p, Q mod q.

Prime Representations Both polynomials mod pa.

Prime-power representations

(n)O(√n) [BBR, Alon]

O(√n) [FW]

All give O(√n) degree symmetric polynomials.

Alon’s ConstructionChoose p ¼ q, and let n = pq -1.

• Let P(X1,…,Xn) = 1 – ( Xi)p-1 mod p

Indicator for Xi being divisible by p.

• Let Q(X1,…,Xn) = 1 – ( Xi)q-1 mod q

Indicator for Xi being divisible by q.

• Both are 1 only for (0,…,0).Degree of the construction is max(p,q) = O(√n).

[BBR’94] Take p = 2, q = 3.Special cases of OR representations modulo pq.

[Frankl-Wilson] Take n = p2 -1. Both polynomials modulo powers of p.

The Ramsey Graph Construction

Ramsey Construction: Vertices: {0,1}n.Edges: Add edge (x,y) if P(x © y) = 0.

Thm: Degree d OR representation gives (G), (G) · nd.

Proof by the linear algebra method [Babai-Frankl].

Plugging in d = O(√n) gives a bound of 2√n.

Lower degree ) better graphs.

Outline of This Talk

I Ramsey graphs from OR representations.• New view of OR representations.• Sample constructions.• Ramsey graphs construction.

II Limitations to Symmetric Constructions.

Limitations to Symmetric Constructions

Thm : (√n) lower bound for symmetric polynomials.

For any OR representation, deg(P) £ deg(Q) = (n).Symmetry vs asymmetry question applies to Ramsey graph constructions.

P mod p, Q mod q. [BBR, Alon]Gives a representation of OR over Zpq.

Known lower bound: √(n/pq).When n < pq [Alon] …

Xi represents OR mod pq.

Both polynomials mod pa. [FW] Based on interpolation algorithm mod pa [G’06].

Thm : (√n) lower bound for symmetric polynomials.

Limitations to Symmetric Constructions

Partition Problem

Adversary gets number n. Picks1. Primes p and q where p¢q > n.2. A µ {1,…, p-1} and B µ {1, …, q-1}Every x 2 {1, …, n} is covered by A or B.Minimize |A|¢|B|.

x mod p lies in A

Partition Problem

Adversary gets number n. Picks1. Primes p and q where p¢q > n.2. A µ {1,…, p-1} and B µ {1, …, q-1}Every x 2 {1, …, n} is covered by A or B.Minimize |A|¢|B|.

1 2 3 4

1 2 3 4 5 6

p = 5, q = 7, n = 12

1 12

…

Partition LemmaTrivial Solutions :

A = {1,…, p-1} and B = {p, 2p, …, }

A = {q, 2q, …} and B = {1, …, q-1}

Gives |A|¢ |B| = n.

Better solutions ) Better OR representations.

Partition Lemma: In any solution, |A|¢|B| ¸ n/8.

Symmetry vs. Asymmetry

Do low degree OR polynomials exist?

Conjecture [Barrington-Beigel-Rudich]: No! (for representations mod 6)• Symmetric polynomials for Symmetric functions. • CRT. Hard explicit construction problem ?

Symmetric polynomials give graphs on {0,1}n based on distances.Q : Are such graphs not good Ramsey graphs?