Foundations of Quasirandomness Joshua N. Cooper UCSD / South Carolina

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

A universe: the class of combinatorial objects OBJ A property: P(o), true a.s. for large objects A sequence: o 1, o 2, o 3, … Define {o i } to be quasirandom if P(o i ) “asymptotically”. A (weak) example: OBJ is the class of graphs, P(G) is the property where as. The rough idea of quasirandomness:

For each random-like property P, one can define P-quasirandomness. Some types of quasirandomness imply other ones: P1P1 P2P2 Q1Q1 P3P3 P4P4 Q2Q2 Q3Q3 By transitivity, the property cliques form a poset: Q1Q1 Q2Q2 Q3Q3 P1P1 P2P2 P3P3 P4P4 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?

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. ?

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 ⊂ X k “Where”: A family L of subsets L ⊂ X k “local sets” Let x be a uniformly random choice of an element from X k, and write 1 R for the indicator of the event that x in R. Then R is quasirandom with respect to L whenever, for all L L,

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

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

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 H d( H )u ° d( H ) k

We wish to reproduce and generalize the theorems appearing in different versions of quasirandomness. For example: Definition. The family L of local sets is called robust if, whenever Y ⊂ X and : Y → L is any mapping, L includes the set Definition. For a set Y ⊂ X k, we write π (Y) for the projection of Y onto the coordinates {2,…,k}. X X X y 1, (y 1 )y 2, (y 2 )y 3, (y 3 )

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} × X k )) is quasirandom with respect to π ( L ). 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. All of the local set systems with k > 1 previously mentioned are robust. (And so is the set of all Cartesion products.)

Current questions (some of which are partially solved): (1) 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.

Thank you!