Title: Foundations of Quasirandomness
1Foundations of Quasirandomness
- Joshua N. Cooper
- UCSD / South Carolina
2Quasirandomness
Regularity
Gowers, Tao 05
Nagle/Rödl/Skokan/Schacht 05
Frankl/Rödl 92, 01
JC 03
JC 05
Permutations
Permutations
3The 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
.
4For 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?
5An 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.
6An 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 family L of 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,
7Intuition 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 L
iff for all L?L.
8Corollary 2. Write Xn. Suppose that
min(R,nk-R) ?(nk). 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 Type
Local Sets
Relations
Graphs / Tournaments
ST, for subsets S,T?V(G)
binary symmetric / antisymmetric
Subsets of Zn
arithmetic progressions (or intervals for weak
quasirandomness)
unary
Permutations
Sets p(I) n J for intervals I, J
binary (inversions)
k-uniform hypergraphs
closed k-uniform hypergraphs
totally symmetric k-ary
9Definition. 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 )
10We wish to reproduce and generalize the theorems
appearing in different versions of
quasirandomness. For example
Definition. For a set Y ? Xk, we write p(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
11Theorem 3. Let k gt 1. If R is quasirandom with
respect to L and L is robust, then, for almost
all x ? X, p(Rn(xXk)) is quasirandom with
respect to p(L).
All of the local set systems with k gt 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.
12Current 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 gt 2?
(5) Describe the structure of the poset of
property cliques induced by the possible
families of local sets.
13Thank you!