Foundations of
Quasirandomness
Joshua N. Cooper
UCSD / South Carolina
Quasirandomness
Regularity
Chung, Graham, Wilson ‘89
Graphs
Szemerédi ’76ish
Simonovits, Sós ‘91
Graphs
Chung, Graham ’90-92
Gowers, Tao – ‘05
Tournaments,
Subsets of Zn,…
Nagle/Rödl/Skokan/Schacht – ’05
Frankl/Rödl – ‘92, ‘01
Chung ‘91
Chung, Graham ’90-92
Hypergraphs
Chung ‘91
Hypergraphs
Kohayakawa, Rödl, Skokan ’02
Hypergraphs (p≠.5)
JC ’03
Permutations
JC ’05
JC ‘05
Permutations
The rough idea of quasirandomness:
A universe: the class of combinatorial objects OBJ
A property: P(o), true a.s. for large objects
o  OBJ
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
1  o(1)  | V (G) |

| E (G) | 
 
2
 2 
where o(1)  0 as
| V (G ) |  .
For each random-like property P, one can define P-quasirandomness.
Some types of quasirandomness imply other ones:
P1
Q1
P3
Q2
P2
P4
Q3
By transitivity, the property cliques form a poset:
P1
Q1
P3
Q2
P2
P4
Q3
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
“Where”: A family
R⊂Xk
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,
PL H1R | L  H1R   o1
PL H1R | L  H1R   o1
Intuition: The statement only has force when
Intuition: Learning that x
P(L) is “not too small”, i.e., (1).
L (“where x is”) tells you almost nothing new about
the event R(x).
Theorem 1. Suppose that min(P(R),1-P(R))
random with respect to
= (1).
L iff
PR  L  P( R)P( L)  o1
for all L
L.
Then R is quasi-
Corollary 2. Write |X|=n. Suppose that min(|R|,nk-|R|)
= (nk).
Then
R is quasirandom with respect to L iff
R L 
for all L
L.
RL
n
k
 
 o nk
… 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
k-uniform
hypergraphs
Sets π(I) ∩ J for intervals I,
J
“closed” k-uniform hypergraphs
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 Kk(k-1) in H
H
d(H )
u ° d(H )
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}.
L of local sets is called robust if, whenever Y ⊂ X
: Y → L is any mapping, L includes the set
Definition. The family
and
y  y .
yY
X
X
X
y1, (y1)
y2, (y2)
y3, (y3)
 y   y 
3
j 1
j
j
L and L is robust,
X, π(R∩({x}×Xk)) is quasirandom with respect to π(L).
Theorem 3. Let k > 1. If R is quasirandom with respect to
then, for almost all x
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
quasirandom.
H is quasirandom, then almost all vertex links are
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!