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Foundations of Quasirandomness

Foundations of Quasirandomness. Joshua N. Cooper UCSD / South Carolina. Chung, Graham, Wilson ‘89. Szemerédi ’76ish. Simonovits, Sós ‘91. Graphs. Graphs. Chung, Graham ’90-92. Tournaments,. Subsets of Z n ,…. Chung, Graham ’90-92. Chung ‘91. Hypergraphs. Chung ‘91.

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Foundations of Quasirandomness

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  1. Foundations of Quasirandomness Joshua N. Cooper UCSD / South Carolina

  2. Chung, Graham, Wilson ‘89 Szemerédi ’76ish Simonovits, Sós ‘91 Graphs Graphs Chung, Graham ’90-92 Tournaments, Subsets of Zn,… Chung, Graham ’90-92 Chung ‘91 Hypergraphs Chung ‘91 Hypergraphs Kohayakawa, Rödl, Skokan ’02 Hypergraphs (p≠.5) JC ‘05 Quasirandomness Regularity Gowers, Tao – ‘05 Nagle/Rödl/Skokan/Schacht – ’05 Frankl/Rödl – ‘92, ‘01 JC ’03 JC ’05 Permutations Permutations

  3. The rough idea of quasirandomness: A universe: the class of combinatorial objects OBJ A property: P(o), true a.s. for large objects A sequence: o1, o2, o3, … Define {oi} to be quasirandom if P(oi) “asymptotically”. A (weak) example: OBJ is the class of graphs, P(G) is the property where as .

  4. Q1 P1 P3 Q2 P2 Q3 P4 By transitivity, the property cliques form a poset: Q1 P1 P3 Q2 P2 Q3 P4 For each random-like property P, one can define P-quasirandomness. Some types of quasirandomness imply other ones: The quasirandom property cliques studied historically have been surprisingly large, i.e., include a large number of very different random-like properties. Furthermore, many of the cliques look similar, even in different universes OBJ. So what exactly is quasirandomness?

  5. An information theoretic idea: Suppose that we have a space X, and a subset of k points of X… ? … then, X is quasirandom if learning whether or not the points are “related” tells us almost nothing about “where” the points are.

  6. An information theoretic idea: Suppose that we have a space X, and a subset of k points of X… … then, X is quasirandom if learning whether or not the points are “related” tells us almost nothing about “where” the points are. “Related”: A relation R⊂Xk “Where”: A familyLof subsets L⊂Xk “local sets” Let x be a uniformly random choice of an element from Xk, and write 1R for the indicator of the event that x in R. Then R is quasirandom with respect to L whenever, for all LL,

  7. Intuition: The statement only has force when P(L) is “not too small”, i.e., (1). Intuition: Learning that xL(“where x is”) tells you almost nothing new about the event R(x). Theorem 1. Suppose that min(P(R),1-P(R)) = (1). Then R is quasi- random with respect to Liff for all LL.

  8. Corollary 2. Write |X|=n. Suppose that min(|R|,nk-|R|) = (nk). Then R is quasirandom with respect to Liff for all LL. … which is why we recover quasirandomness in all its guises when we set: Object Type Local Sets Relations Graphs / Tournaments S×T, for subsets S,T⊂V(G) binary symmetric / antisymmetric Subsets of Zn arithmetic progressions (or intervals for “weak” quasirandomness) unary Permutations Sets π(I)∩J for intervals I, J binary (inversions) k-uniform hypergraphs “closed” k-uniform hypergraphs totally symmetric k-ary

  9. Definition. A k-uniform hypergraph is called “closed” when it is equal to its image under the closure operator u° d, where d(H )= the set of all (k-1)-edges contained in edges of H u(H )= the set of all k-edges spanned by a K(k-1)in H k H d(H ) u° d(H )

  10. X X X y1, (y1) y2, (y2) y3, (y3) We wish to reproduce and generalize the theorems appearing in different versions of quasirandomness. For example: Definition. For a set Y ⊂ Xk, we write π(Y) for the projection of Y onto the coordinates {2,…,k}. Definition. The family L of local sets is called robust if, whenever Y ⊂ X and: Y →L is any mapping, L includes the set

  11. Theorem 3. Let k > 1. If R is quasirandom with respect to L and L is robust, then, for almost all xX, π(R∩({x}×Xk)) is quasirandom with respect to π(L). All of the local set systems with k > 1 previously mentioned are robust. (And so is the set of all Cartesion products.) Translation into two sample contexts: Corollary 4. If a tournament T is quasirandom, then almost all out-degrees are n/2 + o(n). Corollary 5. If a hypergraph H is quasirandom, then almost all vertex links are quasirandom.

  12. Current questions (some of which are partially solved): • What are the conditions on R sufficient to prove the • converse of the theorem on the previous slide? (2) What about substructure counts, i.e., “patterns”? (3) What role does a group structure on X play? (4) Is there a spectral aspect of quasirandomness that goes beyond what is already known? Is it possible to make sense of this question for k > 2? (5) Describe the structure of the poset of property cliques induced by the possible families of local sets.

  13. Thank you!

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