On weakly intersecting pairs of sets Zoltn Kirly 1 Zoltn L. Nagy 1 - - PowerPoint PPT Presentation

Intersecting pairs of sets A new lower bound Summary

On weakly intersecting pairs of sets

Zoltán Király1 Zoltán L. Nagy1 Dömötör Pálvölgyi1,2 Mirkó Visontai3

1Eötvös University Budapest 2Ecole Polytechnique Fédérale de Lausanne 3University of Pennsylvania

July 7, 2010/LPCA

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary

Outline

1

Intersecting pairs of sets Definitions History Previous results

2

A new lower bound A lattice path construction Counting lattice paths

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Outline

1

Intersecting pairs of sets Definitions History Previous results

2

A new lower bound A lattice path construction Counting lattice paths

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Pairs of sets

Definition (An (a, b)-set system) Let F be a family of pairs of sets. We call F an (a, b)-set system if for (A, B) ∈ F we have that |A| = a, |B| = b and A ∩ B = ∅.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Pairs of sets

Definition (An (a, b)-set system) Let F be a family of pairs of sets. We call F an (a, b)-set system if for (A, B) ∈ F we have that |A| = a, |B| = b and A ∩ B = ∅. Example F =

• ({1, 2}, {3, 4, 5}), ({1, 3}, {4, 5, 6})
• is a (2, 3)-set

system.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Intersecting pairs of sets

Definition (Intersecting pairs of sets) An (a, b)-set system F is intersecting if for any (Ai, Bi), (Aj, Bj) ∈ F with i = j we have that Ai ∩ Bj = ∅ and Aj ∩ Bi = ∅.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Intersecting pairs of sets

Definition (Intersecting pairs of sets) An (a, b)-set system F is intersecting if for any (Ai, Bi), (Aj, Bj) ∈ F with i = j we have that Ai ∩ Bj = ∅ and Aj ∩ Bi = ∅. Example F1 =

• ({1, 2}, {3, 4, 5}), ({1, 3}, {4, 5, 6})
• is not intersecting,

but F2 =

• ({1, 2}, {3, 4, 5}), ({3, 9}, {2, 5, 8})
• is.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Outline

1

Intersecting pairs of sets Definitions History Previous results

2

A new lower bound A lattice path construction Counting lattice paths

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Classical results

Theorem (B. Bollobás ’65) The maximum possible size of an intersecting (a, b)-set system is a+b

b

• and this is sharp.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Classical results

Theorem (B. Bollobás ’65) The maximum possible size of an intersecting (a, b)-set system is a+b

b

• and this is sharp.

Theorem (P . Frankl ’82) If we require that Ai ∩ Bj = ∅ for i > j only, then the same holds. Namely, the maximum size of such a set system is a+b

b

• .

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Another relaxation of the problem

The following relaxation was introduced and studied by Tuza. Definition (Weakly intersecting pairs of sets) An (a, b)-set system F is weakly intersecting if for any (Ai, Bi), (Aj, Bj) ∈ F with i = j we have that Ai ∩ Bj and Aj ∩ Bi are not both empty.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Another relaxation of the problem

The following relaxation was introduced and studied by Tuza. Definition (Weakly intersecting pairs of sets) An (a, b)-set system F is weakly intersecting if for any (Ai, Bi), (Aj, Bj) ∈ F with i = j we have that Ai ∩ Bj and Aj ∩ Bi are not both empty. The same upper bound does not hold any more!

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

The objective

Definition Let g(a, b) denote the maximum possible size of a weakly intersecting system.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

The objective

Definition Let g(a, b) denote the maximum possible size of a weakly intersecting system. Problem: Investigate the ratio g(a, b)/ a+b

a

• .

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

The objective

Definition Let g(a, b) denote the maximum possible size of a weakly intersecting system. Problem: Investigate the ratio g(a, b)/ a+b

a

• .

We will show that: lim inf

a+b→∞ g(a, b)/

a + b a

• ≥ 2 − o(1)

.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Outline

1

Intersecting pairs of sets Definitions History Previous results

2

A new lower bound A lattice path construction Counting lattice paths

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Lower bounds by Tuza

Claim 1. g(a, 1) ≥ 2a + 1.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Lower bounds by Tuza

Claim 1. g(a, 1) ≥ 2a + 1. Example

• ({1, 2}, 3}), ({2, 3}, 4), ({3, 4}, 5), ({4, 5}, 1), ({5, 1}, 2)
• .

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Lower bounds by Tuza

Claim 1. g(a, 1) ≥ 2a + 1. Example

• ({1, 2}, 3}), ({2, 3}, 4), ({3, 4}, 5), ({4, 5}, 1), ({5, 1}, 2)
• .

Alternatively: (0, 0, 1, _, _), (_, 0, 0, 1, _), (_, _, 0, 0, 1), (1, _, _, 0, 0), (0, 1, _, _, 0).

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Lower bounds by Tuza

Claim 2. g(a, b) ≥ g(a − 1, b) + g(a, b − 1).

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Lower bounds by Tuza

Claim 2. g(a, b) ≥ g(a − 1, b) + g(a, b − 1). Example

• (1, {2, 3}), (2, {1, 3}), (3, {1, 2})
• ({1, 2}, 3), ({1, 3}, 2), ({2, 3}, 1)
• Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai

On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Lower bounds by Tuza

Claim 2. g(a, b) ≥ g(a − 1, b) + g(a, b − 1). Example

• ({1, 4}, {2, 3}), ({2, 4}, {1, 3}), ({3, 4}, {1, 2})
• ({1, 2}, {3, 4}), ({1, 3}, {2, 4}), ({2, 3}, {1, 4})
• Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai

On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Lower bounds by Tuza

Claim 2. g(a, b) ≥ g(a − 1, b) + g(a, b − 1). Example

• ({1, 4}, {2, 3}), ({2, 4}, {1, 3}), ({3, 4}, {1, 2})
• ({1, 2}, {3, 4}), ({1, 3}, {2, 4}), ({2, 3}, {1, 4})
• Alternatively:

(0, 1, 1), (1, 0, 1), (1, 1, 0), (1, 0, 0), (0, 1, 0), (0, 0, 1) becomes (0, 1, 1, 0), (1, 0, 1, 0), (1, 1, 0, 0), (1, 0, 0, 1), (0, 1, 0, 1), (0, 0, 1, 1).

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Lower bounds by Tuza

Claim 1. g(a, 1) ≥ 2a + 1. Claim 2. g(a, b) ≥ g(a − 1, b) + g(a, b − 1). Corollary g(a, b) ≥ 2 a + b a

• −

a + b − 2 a − 1

• =
• 2 −

ab (a + b)(a + b − 1) a + b a

• .

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary Definitions History Previous results

Lower bounds by Tuza

Claim 1. g(a, 1) ≥ 2a + 1. Claim 2. g(a, b) ≥ g(a − 1, b) + g(a, b − 1). Corollary g(a, b) ≥ 2 a + b a

• −

a + b − 2 a − 1

• =
• 2 −

ab (a + b)(a + b − 1) a + b a

• .

This implies lima→∞ ˜ g(a, a)/ 2a

a

• = 7

4, where ˜

g(a, b) is the RHS.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

New lower bound

We will show that lim inf

a+b→∞ g(a, b)/

a + b a

• ≥ 2 − o(1).

(Whenever we use o(1) we mean that the number tends to 0 as a + b tends to infinity.)

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

Outline

1

Intersecting pairs of sets Definitions History Previous results

2

A new lower bound A lattice path construction Counting lattice paths

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

A lattice path construction

Consider a generalized Dyck path, i.e. a path that starts from (0, 0) and ends at (a, b), each step is either an up step or a right step, all points visited are below the diagonal (of slope b/a). Denote the set of all such paths by L(a, b).

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

Generalized Dyck paths

✧✧✧✧✧✧✧✧✧✧✧✧✧✧✧ ✲ ✲ ✲ ✻ ✲ ✲ ✻ ✻ Figure: A generalized Dyck path from (0, 0) to (5, 3).

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

A property of the generalized Dyck paths

Prefix-suffix property Let π, σ ∈ L(a, b) then no prefix of π can equal a suffix of σ.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

A property of the generalized Dyck paths

Prefix-suffix property Let π, σ ∈ L(a, b) then no prefix of π can equal a suffix of σ. Proof. slope(prefix(π)) < b/a < slope(suffix(σ)).

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

The construction

Idea Take π = (π1, . . . , πa+b) ∈ L(a, b), pad it with a + b − 1 ’_’s and look at all of its cyclic shifts. (If the total length were more than 2a + 2b − 1 then the weak intersection assumption would be violated.) Repeat this for all π ∈ L(a, b).

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

The construction

Example L(2, 3) = {(0, 0, 1, 1, 1), (0, 1, 0, 1, 1)}. Therefore, we get 2 × (5 + 4) = 18 elements in the family as follows: (0, 0, 1, 1, 1, _, _, _, _) (_, 0, 0, 1, 1, 1, _, _, _) . . . (0, 1, 1, 1, _, _, _, _, 0) (0, 1, 0, 1, 1, _, _, _, _) (_, 0, 1, 0, 1, 1, _, _, _) . . . (1, 0, 1, 1, _, _, _, _, 0)

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

The construction

✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✲ ✲ ✻ ✻ ✻ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✡ ✲ ✻ ✲ ✻ ✻ Figure: The two elements of L(2, 3).

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

Outline

1

Intersecting pairs of sets Definitions History Previous results

2

A new lower bound A lattice path construction Counting lattice paths

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

Counting lattice paths

Theorem (Bizley ’54) |L(a, b)| = a + b a

• /(a + b)

if gcd(a, b) = 1.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

Counting lattice paths

Theorem (Bizley ’54) |L(a, b)| = a + b a

• /(a + b)

if gcd(a, b) = 1.

✓ ✓ ✼❩❩ ⑦❩❩ ⑦❩❩ ⑦✓ ✓ ✓ ✼ ✓ ✓ ✼❩❩ ⑦ ❩❩ ⑦❩❩ ⑦❩❩ ⑦✓ ✓ ✼ ✓ ✓ ✼❩❩ ⑦✓ ✓ ✼ t t

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

Computing the lower bound

Theorem g(a, b) ≥

• 2 −

1 a + b a + b a

• ,

if gcd(a, b) = 1.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

Computing the lower bound

Theorem g(a, b) ≥

• 2 −

1 a + b a + b a

• ,

if gcd(a, b) = 1. Proof. g(a, b) ≥ (2a+2b−1)|L(a, b)| = (2a+2b−1) a+b

a

• /(a+b).

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

Computing the lower bound

Corollary g(a, a − 1) ≥

• 2 −

1 2a − 1 2a − 1 a

• .

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

Computing the lower bound

Corollary g(a, a − 1) ≥

• 2 −

1 2a − 1 2a − 1 a

• .

Corollary g(a, a) ≥

• 2 −

1 2a − 1 2a a

• .

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

Main theorem

Theorem g(a, b) ≥ (2 − o(1)) a + b a

• .

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

Main theorem

Theorem g(a, b) ≥ (2 − o(1)) a + b a

• .

Sketch of the proof. Use the g(a, b) ≥ g(a − 1, b) + g(a, b − 1) inequality until we decrease a + b to a prime p for which we automatically have gcd(p − q, q) = 1 for 0 < q < p and then apply the previous theorem.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary A lattice path construction Counting lattice paths

Upper bounds

The following bounds were established by Tuza: Claim g(a, 1) = 2a + 1. Theorem (Tuza ’87) g(a, b) < (a + b)a+b aabb By the Stirling formula (a+b)a+b

aabb

∼ (2abπ/(a + b))1/2 < (2πb)1/2 if a and b are large.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary

Summary

Improved lower bound to the maximal size of weakly intersecting pairs of sets Key ingredient: generalized Dyck paths Exact number or matching asymptotic upper bound is still

• pen.

Computer verification shows that our lattice path construction is not optimal, g(2, 3) ≥ 19.

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary

Thank you for your attention!

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets A new lower bound Summary

Open problems

Is g(a, a) = 2g(a − 1, a) for all a ≥ 2? Is g(a, b) < 2 a+b

a

• ?

Is g(a, a) = o(22k)?

Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets