The Quasi-Randomness of Hypergraph Cut Properties Asaf Shapira & Raphael Yuster

Background Chung, Graham, and Wilson ’89, Thomason ‘87: 1. Defined the notion of a p-quasi-random graph = A graph that “behaves” like a typical graph generated by G(n,p). 2. Proved that several “natural” properties that hold in G(n,p) whp “force” a graph to be p-quasi-random. Abstract Question: When can we say that a single graph behaves like a random graph? “Concrete” problem: Which graph properties “force” a graph to behave like a “truly” random one.

The CGW Theorem Theorem [CGW ‘89]: Fix any 0<p<1, and let G=(V,E) be a graph on n vertices. The following are equivalent: 1.Any set U  V spans  ½p|U| 2 edges 2.Any set U  V of size ½n spans  ½p|U| 2 edges 3. G contains  ½pn 2 edges and  p 4 n 4 copies of C 4 4.Most pairs u,v have co-degree  p 2 n Definition: A graph that satisfies any (and therefore all) the above properties is p-quasi-random, or just quasi-random. Note: All the above hold whp in G(n,p).

Quasi-Random Properties Definition: Say that a graph property is quasi-random if it is equivalent to the properties in the CGW theorem. “The” Question: Which graph properties are quasi-random? Any (reasonable) property that holds in G(n,p) whp? Example 1: Having  ½pn 2 edges and  p 3 n 3 copies of K 3 is not a quasi-random property. Recall that if we replace K 3 with C 4 we do get a quasi-random property. No! Example 2: Having degrees  pn is not a quasi-random prop. …but having co-degrees  p 2 n is a quasi-random property.

The Chung-Graham Theorem Theorem [Chung-Graham ’89]: 1. Having  ¼ pn 2 edges crossing all cuts of size (½n,½n) is not a quasi-random property. 2. For any 0<  <½, having   (1-  )pn 2 edges crossing all cuts of size (  n,(1-  )n) is a quasi-random property. [CG ‘89] Gave two proofs of (2). One using a counting argument, and another algebraic proof based on the rank of certain intersection matrices. To get (1), take an Independent set on n/2 vertices, a clique on the rest, and connect them with a random graph. [Janson ‘09] Gave another proofs of (2), using graph limits.

Quasi-Random Hypergraphs What is a quasi-random hypergraph? Answer 1: The “obvious” generalization of quasi-random graphs. Every set of vertices has the “correct” edge density. Definition: This is called “weak” quasi-randomness. Why? Because it does not imply certain things that are implied by quasi-randomness in graphs. Answer 2: “Strong” quasi-randomness. Fact: Strong Quasi-Randomness  Weak Quasi-Randomness Notation: “P is Quasi-Random” means P  Weak Quasi-Rand

Our Main Results Theorem 1 [S-Yuster ’09]: 1. If  = (1/k,…,1/k) then P  is not quasi-random. Definition: Let  =(  1,…,  k ) satisfy 0<  i <1 and  I =1. Let P  be the following property of k-uniform hypergraphs: Any (  1 n,…,  k n)-cut has the correct number of edges crossing it. 2. If   (1/k,…,1/k) then P  is quasi-random. Theorem 2 [S-Yuster ’09]: 1. When  = (½,½) the only way a non-quasi random graph can satisfy P  is the “trivial” one. 2. Same result conditionally holds in hypergraphs.

Proof Overview Theorem: If  = (1/k,…,1/k) then P  is not quasi-random. Definition: Let  =(  1,…,  k ) satisfy 0<  i <1 and  I =1. We let P  be the following property of k-uniform hypergraphs: Any (  1 n,…,  k n)-cut has the correct number of edges crossing it. 0 2p p [CG‘89] For k=2: For arbitrary k  2: i vertices k-i vertices 2ip/k

Proof Overview Theorem: If   (1/k,…,1/k) then P  is not quasi-random. Proof (of k=2): It is enough to show that every set of vertices of size  n has the correct edge density. Let A be such a set. p2p2 p1p1 p3p3 Let 0  c , and “re-shuffle” the partition by randomly picking cn vertices from A and (  -c)n vertices from V-A. 1. We know the expected number of edges in the new cut. 2. This expectation is a linear function in p 1, p 2, p Using c  {0, ,  /2} we get 3 linear equations, which have a unique solution p 1 =p 2 =p 3 =p when   1/2. A |A|=  n V-A |V-A|=(1-  )n

Proof Overview Theorem 2: When  = (½,½) the only way a non-quasi random graph can satisfy P  is the “trivial” one. Instead of thinking about graphs, let’s consider the problem of assigning weights to the edges of the complete graph, s.t. for any (n/2,n/2)-cut, the total weight crossing it is p. p The random graph 0 2p p The example showing that P  is not quasi random What is “trivial”? Two ways a graph can satisfy P  There are many such solutions

Proof Overview What is “trivial”? Two ways a graph can satisfy P  p The random graph (1) 0 2p p The example showing that P  is not quasi random (2) Satisfying P  is equivalent to satisfying a set of linear equations: 1.Unknowns are the weights of the edges. 2.We have one linear equation for any (n/2,n/2)-cut Definition: A trivial solution is any affine combination of solution (1) and (a collection of) solution (2).

Proof Overview p The random graph (1) 0 2p p The example showing that P  is not quasi random (2) Definition: A trivial solution is any affine combination of solution (1) and (a collection of) solution (2). Theorem 2: When  = (½,½) the only solutions satisfying P  are the trivial ones.

Proof Overview Definition: A trivial solution is any affine combination of solution (1) and (2). Theorem 2: When  = (½,½) the only solutions satisfying P  are the trivial ones. Proof: Any solution is a solution of the linear system Ax=p. Recall Satisfying P  is equivalent to satisfying a set of linear equations: 1. Unknowns are the weights of the edges. 2. We have one equation for any (n/2,n/2)-cut p The random graph (1) 0 2p p The example showing that P  is not quasi random (2) Step 2: span[solutions (2) – (1)] has dimension  n-1. Step 1: rank(A)  Definition: Let us write this as Ax=p. Note: A is an matrix.

Proof Overview Definition: A trivial solution is any affine combination of solution (1) and (2). p The random graph (1) 0 2p p The example showing that P  is not quasi random (2) Step 2: trivial solutions have dimension  n-1. Proof: For every solution (2) consider the vector of pairs (v 1,v i ). 2p0 p n/2-1 of the entries are 2p, the other are p. After subtracting (1) from these vectors, we get, for every subset S  [n-1] of size n/2, a vector v S, satisfying: 1. v S (i) = 0 if i  S. 2. v S (i) = p if i  S. This collection spans R n-1

Proof Overview Note: A is an matrix n/2 n/2 n/2-1 Proof: Take the vector v S corresponding to some cut. v S,t = vector of cut obtained by moving t from S to V-S. We first prove that matrix of (n/2,n/2-1)-cuts has full rank. (v S,t – v S ) 1  t t (v S,t – v S )  t t vs-vs- S V-S C -c Step 1: rank(A) 

Proof Overview Conclusion: A spans the rows of the matrix I (2,n/2,n-1) n/2-element subsets of [n-1] 2-element subsets of [n-1] I S,T = 1 iif S  T [Gottlieb ‘66]: rank( I (2,h,k)) =. Step 1: rank(A) 

Concluding Remarks Coro: If   (1/3,1/3,1/3) then P  is quasi-random Definition: Let  =(  1,  2,  3 ) satisfy 0<  i <1 and  i =1. Let P  be the following graph property: Any (  1 n,  2 n,  3 n)-cut is crossed by the “correct” number of K 3. Open Problem: What happens when  = (1/3,1/3,1/3)? Proof: Replace every K 3 with a 3-hyper edge. We get a hypergraph satisfying P , which must be quasi-random by Theorem 1. This means that in the graph, any set of vertices has the “correct” number of K 3. A theorem of Simonovits-Sos implies that the graph must be quasi-random.

Thank You

Background Relation to (theoretical) computer science: 1. Conditions of randomness that are verifiable in polynomial time. For example, using number of C 4, or using 2 (G). 2. Algorithmic version of Szemeredi’s regularity-lemma: [Alon et al. ’95] Uses equivalence between quasi-randomenss and co-degrees.

Background Relation to Extremal Combinatorics: 1. Central in the strong hypergraph generalizations of Szemeredi’s regularity-lemma [RSSN’04, Gowers’06, Tao’06]. Quasi-Random Groups [Gowers ‘07] Generalized Quasi-Random Graphs [Lovasz-Sos ‘06] Quasi-Random Set Systems [Chung-Graham ‘91]