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Parikshit Gopalan Georgia Institute of Technology Atlanta, Georgia, USA.

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Explicit Ramsey Graph Constructions [Erdös] : There exists a graph G on 2 n vertices with (G), (G) · 2n. Probabilistic Method. $100 for explicit construction. [Ramsey] : Every graph on 2 n vertices has either an independent set or a clique of size n/2. Easy to construct G on 2 n vertices with (G), (G) · 2 n/2.

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Alternate View Constructing Ramsey graphs: Two color the edges of K n so that there are no large monochromatic cliques. Constructing Multicolor Ramsey graphs: Color the edges of K n using t colors so that there are no large monochromatic cliques.

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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) · 2 n. Extractors and pseudorandomness.

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Lower Bounds for AC 0 [m] Problem: Find a function that cannot be computed by polynomial size, constant depth circuits with AND, OR and Mod-m gates. For Mod-p gates (p is prime) [Razborov, Smolensky] With p = 2, the Mod-3 function is hard. For Mod-m gates (m is composite, say 6) With m = 6, is the Mod-5 function hard ? No lower bounds known for any natural function.

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Polynomial Representations of Boolean functions Def: P(X 1,…,X n ) over Z m represents f: {0,1} n ! {0,1} if f(x) f(y) ) P(x) P(y) mod m Lower bounds for AC 0 [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 Z m. (Simpler problem?)

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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(n 1/2 ) 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]

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A Connection [Grolmusz] Problem: Let F be a family of subsets S i of [n] where |S i | = 0 mod m |S i Å S j | 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.

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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.

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Outline of This Talk I Ramsey graphs from OR representations. New view of OR representations. Sample constructions. Ramsey graphs. II Limitations to Symmetric Constructions.

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Outline of This Talk I Ramsey graphs from OR representations. New view of OR representations. Sample constructions. Ramsey graphs. II Limitations to Symmetric Constructions.

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OR Representations New view of an OR representation: Two polynomials s.t. the union of their zero sets is {0,1} n \ {0}.

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P = 0Q = 0

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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 p a. Prime-power representations (n) O(n) [BBR, Alon] O(n) [FW] All give O(n) degree symmetric polynomials.

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Alons Construction Choose p ¼ q, and let n = pq -1. Let P(X 1,…,X n ) = 1 – ( X i ) p-1 mod p Indicator for X i being divisible by p. Let Q(X 1,…,X n ) = 1 – ( X i ) q-1 mod q Indicator for X i being divisible by q. Both are 1 only for (0,…,0). Degree of the construction is max(p,q) = O(n). [BBR94] Take p = 2, q = 3. Special cases of OR representations modulo pq. [Frankl-Wilson] Take n = p 2 -1. Both polynomials modulo powers of p.

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The Frankl-Wilson Construction Let n = p 2 -1. P(X 1,…,X n ) stays the same. Indicator for X i being divisible by p. Remaining weights are 0, p, 2p, …, p 2 -p. Q(X 1, …, X n ) = ( X i –p)...( X i – p 2 + p)mod p p Both are 1 only for (0,…,0). Degree of the construction is p = O(n). Does not use Chinese Remaindering. Extends to multicolor graph constructions.

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Plug in X = x 1 : P(0,…,0) 0, P(x 1 © x 2 ) = 0, …, P(x 1 © x k ) = 0. 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) · n d. Consider a clique: x 1, …, x k We have: P(x i © x j ) = 0. Claim : Polynomials P(X © x 1 ), …, P(X © x k ) are LI. Dimension of vector space O(n d ). Hence k · n d. Bound size of (G) using Q(X 1,…,X n ).

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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) · n d. Proof by the linear algebra method [Babai-Frankl]. Plugging in d = O(n) gives a bound of 2 n. Lower degree ) better graphs.

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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.

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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.

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P mod p, Q mod q. [BBR, Alon] Gives a representation of OR over Z pq. Known lower bound: (n/pq). When n < pq [Alon] … X i represents OR mod pq. Both polynomials mod p a.[FW] Based on interpolation algorithm mod p a [G06]. Thm : (n) lower bound for symmetric polynomials. Limitations to Symmetric Constructions

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High-Level Idea 1.Algebraic Step: Characterize zero-sets of low- degree multivariate polynomials. 2.Combinatorial Step: Show that there is no good partition of {0,1} n. Symmetry : Replace multivariate by univariate. Replace {0,1} n by {0,…,n}

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Partition Problem Adversary gets number n. Picks 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

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Partition Problem Adversary gets number n. Picks 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|. 1 2 3 4 1 2 3 4 5 6 p = 5, q = 7, n = 12 112…

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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.

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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?

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Prime-Power Representations Algebraic Step : Given a univariate polynomial P(X) mod p a which is 0 at ( 1, …, k ). Give a lower bound on deg(P). Low degree polynomials mod p a have many roots. Eg: X 3 = 0 mod 27 (0,3,6 …, 24) 9 X = 0 mod 27(0,3,6 …, 24) Structure theorem for zero sets of low-degree polynomials. (Builds on interpolation algorithm [G06])

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603 mod 9 01 2 mod 3 09121114 Roots of low degree polynomials are clustered together. Let m = 27. I Zero-Sets of low degree polynomials

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