Multicolor Hypergraph Containers Victor Falgas-Ravry 1 Kelly OConnell - - PowerPoint PPT Presentation

Background and previous results Our results Summary

Multicolor Hypergraph Containers

Victor Falgas-Ravry1 Kelly O’Connell2 Johanna Strömberg3 Andrew J. Uzzell4

1Umeå University 2Vanderbilt University 3Uppsala University 4University of Nebraska–Lincoln

May 21, 2016

Background and previous results Our results Summary

Overview

Today, I will talk about: the hypergraph container method and its applications; hereditary properties of graphs and multicolored graphs (including oriented and directed graphs); how to use hypergraph containers to determine the number

• f multicolored graphs that satisfy a given hereditary property.

Background and previous results Our results Summary

Outline

1

Background and previous results Hypergraph containers and their applications Hereditary properties of graphs

2

Our results Hereditary properties of multicolored graphs Multicolor hypergraph containers and their applications

Background and previous results Our results Summary

Outline

1

Background and previous results Hypergraph containers and their applications Hereditary properties of graphs

2

Our results Hereditary properties of multicolored graphs Multicolor hypergraph containers and their applications

Background and previous results Our results Summary

Outline

1

Background and previous results Hypergraph containers and their applications Hereditary properties of graphs

2

Our results Hereditary properties of multicolored graphs Multicolor hypergraph containers and their applications

Background and previous results Our results Summary

Definitions and notation: H-free (hyper)graphs

Definition Let H be a graph. We say that a graph G is H-free if H is not a subgraph of G. (If H is a hypergraph, the definition of H-free is the same.) Definition Let H be a graph. The Turán number of H, denoted ex(n,H), is the maximum number of edges in an H-free graph on n vertices.

Background and previous results Our results Summary

Definitions and notation: hypergraphs and independent sets

Definitions Let H be a hypergraph. We say that I ⊆ V (H) is an independent set if no edge of H is contained entirely in I. α(H) denotes the maximum size of an independent set in H. I(H) denotes the family of independent sets in H i(H) := |I(H)| denotes the number of independent sets in H. Definition An r-uniform hypergraph (or r-graph) is a hypergraph in which each edge contains exactly r vertices.

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The hypergraph G(n,K3)

Definition Given n, let G(n,K3) be the 3-graph with vertex set E(Kn) such that {e1,e2,e3} ∈ E(G(n,K3)) if and only if {e1,e2,e3}, considered as a graph, forms a copy of K3.

a b c e f d a d b e c f

Background and previous results Our results Summary

Two well-known theorems concerning G(n,K3)

Definition Given n, let G(n,K3) be the 3-graph with vertex set E(Kn) such that {e1,e2,e3} ∈ E(G(n,K3)) if and only if {e1,e2,e3}, considered as a graph, forms a copy of K3. Theorem ( ) α

• G(n,K3)
• =

n2 4

• .

Theorem ( ) i

• G(n,K3)
• = 2

n2 4 +o(n2).

Background and previous results Our results Summary

Two well-known theorems concerning G(n,K3)

Definition Given n, let G(n,K3) be the 3-graph with vertex set E(Kn) such that {e1,e2,e3} ∈ E(G(n,K3)) if and only if {e1,e2,e3}, considered as a graph, forms a copy of K3. Theorem (Mantel, 1907) α

• G(n,K3)
• =

n2 4

• .

Theorem (Erdős, Kleitman, & Rothschild, 1976) i

• G(n,K3)
• = 2

n2 4 +o(n2).

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Well-known theorems reinterpreted

Observations There is a 1-to-1 correspondence between triangle-free graphs

• n [n] and independent sets in G(n,K3).

In particular, ex(n,K3) = α(G(n,K3)) and the number of triangle-free graphs on [n] equals i(G(n,K3)).

a b c e f d a d b e c f

Background and previous results Our results Summary

Problems involving independent sets in hypergraphs

We have translated extremal problems into questions about independent sets in hypergraphs. Problem For many hypergraphs H, I(H) is a large, complicated family. Solution Replace I(H) with a smaller, simpler family.

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Hypergraph containers

What we want Given a hypergraph H of order N and ε > 0, we want a collection C

• f subsets of V (H) such that:

1 C is a container family for I(H). 2 C is small. 3 Elements of C are almost independent.

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Hypergraph containers

What we want Given a hypergraph H of order N and ε > 0, we want a collection C

• f subsets of V (H) such that:

1 C is a container family for I(H). (Every I ∈ I(H) is a subset of

some C ∈ C.)

2 C is small. (|C| ≤ 2εN.) 3 Elements of C are almost independent. (For all C ∈ C,

e(H[C]) ≤ εe(H).)

Background and previous results Our results Summary

Hypergraph containers

What we want Given a hypergraph H of order N and ε > 0, we want a collection C

• f subsets of V (H) such that:

1 C is a container family for I(H). (Every I ∈ I(H) is a subset of

some C ∈ C.)

2 C is small. (|C| ≤ 2εN.) 3 Elements of C are almost independent. (For all C ∈ C,

e(H[C]) ≤ εe(H).) Balogh, Morris, & Samotij (2015) and Saxton & Thomason (2015) independently showed that every hypergraph has such a container family. Graph containers were first used by Kleitman & Winston (1982) and by Sapozhenko (1987–2003).

Background and previous results Our results Summary

A hypergraph container family for G(n,K3)

We count triangle-free graphs by finding a container family for G(n,K3). Theorem (Saxton & Thomason, 2015) Let ε > 0. For all n sufficiently large, there exists a collection C of graphs on [n] such that

1 Every triangle-free graph on [n] is a subgraph of some C ∈ C. 2 |C| ≤ 2εn2. 3 If C ∈ C, then C contains at most εn3 triangles and

e(C) ≤ 1

2 +ε

n

2

• .

Background and previous results Our results Summary

A hypergraph container family for G(n,K3)

We count triangle-free graphs by finding a container family for G(n,K3). Theorem (Saxton & Thomason, 2015) Let ε > 0. For all n sufficiently large, there exists a collection C of graphs on [n] such that

1 Every triangle-free graph on [n] is a subgraph of some C ∈ C. 2 |C| ≤ 2εn2. 3 If C ∈ C, then C contains at most εn3 triangles and

e(C) ≤ 1

2 +ε

n

2

• .

Remark: By a “supersaturation” result of Erdős and Simonovits, if a graph C has at most εn3 triangles, then e(C) ≤ 1

2 +ε

n

2

• .

Applications of the container method nearly always rely on supersaturation results.

Background and previous results Our results Summary

Using hypergraph containers to count triangle-free graphs

Theorem (Saxton & Thomason, 2015) Let ε > 0. For all n sufficiently large, there exists a collection C of graphs on [n] such that

1 Every triangle-free graph on [n] is a subgraph of some C ∈ C. 2 |C| ≤ 2εn2. 3 If C ∈ C, then C contains at most εn3 triangles and

e(C) ≤ 1

2 +ε

n

2

• .

Let Pn denote the class of triangle-free graphs on [n]. Every C ∈ C has at most 2e(C) triangle-free subgraphs. The theorem then implies that |Pn| ≤ |C|·2maxC∈C e(C) ≤ 2εn22

n2 4 +εn2 = 2 n2 4 +o(n2).

For the lower bound, take all 2n2/4 subgraphs of Kn/2,n/2.

Background and previous results Our results Summary

Applications of hypergraph containers: counting

(Hyper)graph containers have been used to count a variety of discrete objects: H-free (hyper)graphs of order n (Kleitman & Winston (1982); Balogh & Samotij (2011, 2011); Balogh, Morris, & Samotij (2015); Saxton & Thomason (2015); Morris & Saxton (2016+)) Antichains in the Boolean lattice (Balogh, Treglown, & Wagner (2016+)) Discrete metric spaces with a specified number of distances (Balogh & Wagner (2016)) t-error-correcting binary codes (Balogh, Treglown, & Wagner (2016+)) Sum-free subsets of [n] and of abelian groups (Sapozhenko (2003); Alon, Balogh, Morris, & Samotij (2014)) Matroids on n elements (Bansal, Pendavingh, & van der Pol (2015))

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Applications of hypergraph containers: typical structure

Containers can also be used to characterize the typical structure of discrete objects: H-free (hyper)graphs (Balogh, Morris, & Samotij (2015)) Edge-maximal triangle-free graphs (Balogh, Liu, Petříčková, & Sharifzadeh (2015)) Graphs without cliques of order r, where r grows slowly with n (Balogh, Bushaw, Collares Neto, Liu, Morris, & Sharifzadeh (2016+)) Graphs without induced even cycles (Kim, Kühn, Osthus, & Townsend (2015+))

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Applications of hypergraph containers: transference I

Finally, containers can be used to “transfer” results about dense structures to sparse random settings. Here is one example. Theorem (Erdős–Stone–Simonovits (1946, 1966)) Let H be a graph and let ε > 0. For all n sufficiently large, if F is an H-free graph of order n, then e(F) ≤

• 1−

1 χ(H)−1 +ε n 2

• .

Definition We say that a graph G is (H,ε)-Turán if the largest H-free subgraph F of G satisfies e(F) ≤

• 1−

1 χ(H)−1 +ε

• e(G).

Background and previous results Our results Summary

Applications of hypergraph containers: transference II

The Erdős–Stone–Simonovits theorem says that for all H and ε, if n is sufficiently large, then Kn is (H,ε)-Turán. What about sparse graphs?

Background and previous results Our results Summary

Applications of hypergraph containers: transference II

The Erdős–Stone–Simonovits theorem says that for all H and ε, if n is sufficiently large, then Kn is (H,ε)-Turán. What about sparse graphs? Definition Given p = p(n) ∈ [0,1], an Erdős–Rényi random graph G(n,p) is a graph on n vertices in which edges are present independently with probability p. Question Given a graph H and ε > 0, for which p is G(n,p) (H,ε)-Turán with high probability?

Background and previous results Our results Summary

Applications of hypergraph containers: transference III

Observation With high probability, G(n,p) has (1+o(1))p n

2

• edges.

Theorem (Schacht, 2009+; Conlon & Gowers, 2010+; Balogh, Morris, & Samotij, 2015; Saxton & Thomason, 2015) Let H be a graph. There exists a constant cH > 0 such that for all ε > 0, there exists a constant c > 0 such that if p ≥ cn−cH, then with high probability the largest H-free subgraph F of G(n,p) satisfies e(F) ≤

• 1−

1 χ(H)−1 +ε

• p

n 2

• ,

i.e., with high probability, G(n,p) is (H,ε)-Turán. Hypergraph containers also give proofs of sparse random versions of Szemerédi’s theorem, Ramsey’s theorem, and Sperner’s theorem.

Background and previous results Our results Summary

Outline

1

Background and previous results Hypergraph containers and their applications Hereditary properties of graphs

2

Our results Hereditary properties of multicolored graphs Multicolor hypergraph containers and their applications

Background and previous results Our results Summary

Hereditary properties of graphs

Definition A graph property P is hereditary if it is closed under removal of vertices (equivalently, under taking induced subgraphs). Examples of hereditary graph properties H-free graphs r-partite graphs Perfect graphs Chordal graphs Observation Every hereditary graph property is defined by a family of forbidden induced subgraphs.

Background and previous results Our results Summary

The speed of a hereditary graph property

Definition Given a graph property P, let Pn = {G ∈ P : |V (G)| = n}. The function n → |Pn| is called the speed of P. Theorem (Alekseev, 1992; Bollobás & Thomason, 1997) If P is a hereditary graph property, then there exists r = r(P) ∈ N∪{∞} such that |Pn| = 2

• 1− 1

r +o(1)

• (n

2).

Background and previous results Our results Summary

Outline

1

Background and previous results Hypergraph containers and their applications Hereditary properties of graphs

2

Our results Hereditary properties of multicolored graphs Multicolor hypergraph containers and their applications

Background and previous results Our results Summary

Outline

1

Background and previous results Hypergraph containers and their applications Hereditary properties of graphs

2

Our results Hereditary properties of multicolored graphs Multicolor hypergraph containers and their applications

Background and previous results Our results Summary

Multicolored graphs

Observation A k-colored graph on [n] corresponds to a k-coloring of E(Kn). Some examples of k-colored graphs Simple graphs (k = 2): present edges red, absent edges blue. Multigraphs with bounded multiplicity: colors record edge multiplicities. Oriented graphs (k = 3): Let V (G) = [n] and let i < j. We set c(ij) =      1, i ≁ j, 2, i → j, 3, i ← j. Directed graphs (k = 4): Same as for oriented graphs, but with c(ij) = 4 if i ⇄ j.

Background and previous results Our results Summary

Hereditary properties of multicolored graphs

Definition Let P be a class of k-colored graphs. We say that P is hereditary if it is closed under removing vertices. Examples For k = 3, Pn = {3-colorings of E(Kn) with no rainbow K3}. For k = 4, Pn = {4-colorings of E(Kn) with no K3 in color 4}.

Background and previous results Our results Summary

The number of digraphs without forbidden oriented graphs

Let H be an oriented graph. Kühn, Osthus, Townsend, & Zhao (2015+) calculated the asymptotic number of H-free oriented graphs of order n the asymptotic number of H-free directed graphs of order n in terms of a certain “weighted Turán number”.

Background and previous results Our results Summary

The number of digraphs without forbidden oriented graphs

Let H be an oriented graph. Kühn, Osthus, Townsend, & Zhao (2015+) calculated the asymptotic number of H-free oriented graphs of order n the asymptotic number of H-free directed graphs of order n in terms of a certain “weighted Turán number”. If H is a digraph, their methods do not give the asymptotic number

• f H-free digraphs. One example is the “double triangle” DK3.

Question How many DK3-free digraphs are there on n vertices?

Background and previous results Our results Summary

Templates for k-colorings

Definition Given k ≥ 2 and n, a template for a k-coloring of E(Kn) is a map t : E(Kn) → P([k])\ / 0. A coloring c realizes t if c(e) ∈ t(e) for all e ∈ E(Kn).

{1, 3, 4} {1, 4} {1, 2, 3}

t

1 4 3

c

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The entropy of a template

Definition Given a template t, let t denote the set of realizations of t. Clearly,

• t
• =
• e∈E(Kn)

|t(e)|. The entropy of t is Ent(t) := logk

• t
• = logk
• e∈E(Kn)

|t(e)| =

• e∈E(Kn)

logk|t(e)|. Observations Let t be a template for k-colorings. We have 0 ≤ Ent(t) ≤ n

2

• .

The entropy of t measures the “uncertainty” in a uniformly random realization of t.

Background and previous results Our results Summary

The entropy density of a hereditary property

Definition Let P be a hereditary property of k-colored graphs. The extremal entropy of P is ex(n,P) := max

• Ent(t) : t is a template with t ⊆ Pn
• and the entropy density of P is

π(P) := lim

n→∞

ex(n,P) n

2

• .

Background and previous results Our results Summary

The entropy density of a hereditary property

Definition Let P be a hereditary property of k-colored graphs. The extremal entropy of P is ex(n,P) := max

• Ent(t) : t is a template with t ⊆ Pn
• and the entropy density of P is

π(P) := lim

n→∞

ex(n,P) n

2

• .

Remark: Why do we borrow the notation for Turán problems? If k = 2, H is a graph, and P is the class of H-free graphs, then ex(n,P) = ex(n,H) and π(P) = π(H). Much as for the Turán density, the proof that the limit π(P) exists uses an averaging argument.

Background and previous results Our results Summary

The entropy density of the class of DK3-free digraphs

Theorem (Falgas-Ravry, O’Connell, Strömberg, & U., 2016+) If P is the class of DK3-free digraphs, then π(P) = 1

2 + log4 3 2

. Proof sketch: i ⇄ j corresponds to color 4, so DK3 is a triangle in color 4 and, by Mantel, any DK3-free digraph has at most n2/4 edges in color 4. Hence, for all 4-coloring templates t with t ⊆ Pn, Ent(t) ≤ 1· n2 4 +log4 3· n 2

• − n2

4

• ∼ 1+log4 3

2 n 2

• . (1)

The template below shows that (1) is sharp.

[3] [3] [4]

Background and previous results Our results Summary

Outline

1

Background and previous results Hypergraph containers and their applications Hereditary properties of graphs

2

Our results Hereditary properties of multicolored graphs Multicolor hypergraph containers and their applications

Background and previous results Our results Summary

A multicolor container theorem

Theorem (Falgas-Ravry, O’Connell, Strömberg, & U., 2016+) Let P be a hereditary property of k-colored graphs. Fix ε > 0 and m ∈ N. For all n sufficiently large, there exists a family of templates Cn such that:

1 Pn ⊆

t∈Cnt.

2 |Cn| ≤ kε(n 2). 3 If t ∈ Cn and c realizes t, then there are at most ε

n

m

• sets A ⊆ [n] of size m such that c |Kn[A] /

∈ Pm.

4 Every template t ∈ Cn has entropy at most (π(P)+ε)

n

2

• .

Background and previous results Our results Summary

A multicolor container theorem

Theorem (Falgas-Ravry, O’Connell, Strömberg, & U., 2016+) Let P be a hereditary property of k-colored graphs. Fix ε > 0 and m ∈ N. For all n sufficiently large, there exists a family of templates Cn such that:

1 Pn ⊆

t∈Cnt.

2 |Cn| ≤ kε(n 2). 3 If t ∈ Cn and c realizes t, then there are at most ε

n

m

• sets A ⊆ [n] of size m such that c |Kn[A] /

∈ Pm.

4 Every template t ∈ Cn has entropy at most (π(P)+ε)

n

2

• .

Remarks: Item (1) says that Cn is a container family for Pn, (2) says that Cn is small, (3) says that every template t ∈ Cn is “almost” in Pn, and (4) says that every t ∈ Cn is “small”, i.e., |t| is not too large.

Background and previous results Our results Summary

The speed of a hereditary property of multicolored graphs

We use our multicolor container theorem to prove the following: Corollary (Falgas-Ravry, O’Connell, Strömberg, & U., 2016+) If P is a hereditary property of k-colored graphs, then |Pn| = k

• π(P)+o(1)
• (n

2).

Background and previous results Our results Summary

The speed of a hereditary property of multicolored graphs

We use our multicolor container theorem to prove the following: Corollary (Falgas-Ravry, O’Connell, Strömberg, & U., 2016+) If P is a hereditary property of k-colored graphs, then |Pn| = k

• π(P)+o(1)
• (n

2).

Corollary (Falgas-Ravry, O’Connell, Strömberg, & U., 2016+) If P is the class of digraphs with no copy of DK3, then |Pn| = 4

• 1

2+ log4 3 2

+o(1)

• (n

2) = 4

• 1

2 +o(1)

• (n

2)3

• 1

2 +o(1)

• (n

2).

Proof: π(P) = 1

2 + log4 3 2

.

Background and previous results Our results Summary

The speed of hereditary properties: proof sketch I

Corollary (Falgas-Ravry, O’Connell, Strömberg, & U., 2016+) If P is hereditary, then |Pn| = k

• π(P)+o(1)
• (n

2).

Ex.: Let P be the class of 3-colored graphs with no rainbow K3. Form a 3-graph H from k = 3 copies of E(Kn), with {e1,e2,e3} ∈ E(H) if and only {e1,e2,e3} forms a rainbow K3.

a b c e f d a b c

Templates correspond to subsets of V (H), and independent sets correspond to templates t with t ⊆ Pn. N.B.: In general, maximum-entropy templates are not necessarily largest independents set in H, and vice versa.

Background and previous results Our results Summary

The speed of hereditary properties: proof sketch II

Corollary (Falgas-Ravry, O’Connell, Strömberg, & U., 2016+) If P is hereditary, then |Pn| = k

• π(P)+o(1)
• (n

2).

a b c

Apply our container theorem to H to get a small container family Cn for {t : t ⊆ Pn} such that if s ∈ Cn, then almost all realizations of s are in Pn. (I.e., s is “almost independent”.) Use a supersaturation result to conclude that if s ∈ Cn, then Ent(s) ≤ (π(P)+ε) n

2

• .

Hence, |Pn| ≤ |Cn|kmaxt∈Cn Ent(t) ≤ kε(n

2)k

• π(P)+ε
• (n

2).

Background and previous results Our results Summary

An open problem

Theorem (Alekseev, 1992; Bollobás & Thomason, 1997) If P is a hereditary graph property, then there exists r = r(P) ∈ N∪{∞} such that |Pn| = 2

• 1− 1

r +o(1)

• (n

2).

Thus, if P is a hereditary property of 2-colored graphs, then π(P) ∈

• 0, 1

2, 2 3, 3 4,...

• ∪{1}.

Question Fix k ≥ 3. If P is a hereditary property of k-colored graphs, what values can π(P) take?

Background and previous results Our results Summary

Summary

Hypergraph containers are a powerful method for solving problems that can be stated in terms of independent sets in hypergraphs. We used hypergraph containers to determine the asymptotic speeds of hereditary properties of multicolored graphs.

Background and previous results Our results Summary

For further reading

• J. Balogh, R. Morris, and W. Samotij.

Independent sets in hypergraphs. Journal of the American Mathematical Society, 28(3):669–709, 2015.

• W. Samotij.

Counting independent sets in graphs. European Journal of Combinatorics, 48:5–18, 2015.

• D. Saxton and A. Thomason.

Hypergraph containers. Inventiones Mathematicae, 201(3):925–992, 2015.

• D. Saxton and A. Thomason.

Simple containers for simple hypergraphs. Combinatorics, Probability and Computing, 25(3):448–459, 2016.

Appendix

Outline

3

Appendix

Appendix

Graph limits, graphons, and entropy

In the theory of graph limits, certain sequences of graphs are defined to be convergent. The limit of a convergent sequence of graphs can be represented by a graphon, a symmetric, measurable function W : [0,1]2 → [0,1]. Definition For x ∈ (0,1), let h(x) = −x log2 x −(1−x)log2(1−x). Given a graphon W , the entropy of W is Ent(W ) :=

• [0,1]2 h
• W (x,y)
• .

1 2 1 2

1

Ent(W ) = 1

4(1+0+0+1) = 1 2,

where h(0), h(1) := 0.

Appendix

Entropy and the speed of hereditary properties

Definition If P is a graph property, let P denote the set of graphons that represent limits of sequences of graphs that lie in P. Theorem (Hatami, Janson, & Szegedy, 2013+) If P is a hereditary graph property, then lim

n→∞

log2|Pn| n

2

• = max

W ∈ P

Ent(W ).

Appendix

Entropy and the speed of hereditary properties

Definition If P is a graph property, let P denote the set of graphons that represent limits of sequences of graphs that lie in P. Theorem (Hatami, Janson, & Szegedy, 2013+) If P is a hereditary graph property, then lim

n→∞

log2|Pn| n

2

• = max

W ∈ P

Ent(W ). We extend this result to hereditary properties of k-colored graphs. Hatami, Janson, & Szegedy also showed that for any hereditary property P, maxW ∈

P Ent(W ) = 1−1/r for some r ∈ N∪{∞},

which recovers the Alekseev–Bollobás–Thomason theorem.

Appendix

The results of Kühn, Osthus, Townsend, & Zhao I

Definition Given a digraph G, let f1(G) =

• {(i,j) : i → j or i ← j but not both}
• and let

f2(G) =

• {(i,j) : i ⇄ j}
• .

Let e2(G) = 2f2(G)+f1(G). Observe that every digraph G has 4f2(G)2f1(G) = 2e2(G) subdigraphs. Definition Given a digraph H, let exdi(n,H) = max{e2(G) : G is an H-free digraph of order n}.

Appendix

The results of Kühn, Osthus, Townsend, & Zhao II

Definition Given a digraph H, let exdi(n,H) = max{e2(G) : G is an H-free digraph of order n}. Theorem (Kühn, Osthus, Townsend, & Zhao, 2015+) If H is an oriented graph, then there are 2exdi(n,H)+o(n2) H-free digraphs of order n.

Appendix

The results of Kühn, Osthus, Townsend, & Zhao II

Definition Given a digraph H, let exdi(n,H) = max{e2(G) : G is an H-free digraph of order n}. Theorem (Kühn, Osthus, Townsend, & Zhao, 2015+) If H is an oriented graph, then there are 2exdi(n,H)+o(n2) H-free digraphs of order n. Let H = DK3. We have exdi(n,DK3) = n2

2 +2

n/2

2

• , but we have

seen that there are at least 4

n2 4 32(n/2 2 ) ≫ 2 n2 2 +2(n/2 2 )

DK3-free digraphs of order n.