### 0 - Microsoft Research

```Constructing Ramsey Graphs
from
Boolean Function Representations.
Parikshit Gopalan
Georgia Institute of Technology
Atlanta, Georgia, USA.
Explicit Ramsey Graph Constructions
[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.
[Erdös] : There exists a graph G on 2n
vertices with (G), (G) · 2n.
Probabilistic Method.
\$100 for explicit construction.
Alternate View
Constructing 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.
n
[Barak-Rao-Shaltiel-Wigderson] : (G), (G) · 2 .
• 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 circuits
Degree 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]
[Barrington-Beigel-Rudich, Grolmusz, Tsai, BarringtonTardos, Green, Alon-Beigel, Bhatnagar-G.-Lipton, Hansen]
Can asymmetry help compute a symmetric function?
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.
(n)
 P mod p, Q mod q.
O(√n) [BBR, Alon]
Prime Representations
 Both polynomials mod pa.
O(√n) [FW]
Prime-power representations
All give O(√n) degree symmetric polynomials.
Alon’s Construction
Choose 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.
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.
Limitations to Symmetric Constructions
Thm : (√n) lower bound for symmetric polynomials.

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].
Partition Problem
1. 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
1. 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|.
p = 5, q = 7, n = 12
1 2 3 4
1
1
…
2 3 4 5 6
12
Partition Lemma
Trivial 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?
```