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?