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 Király 1 Zoltán L. Nagy 1 Dömötör Pálvölgyi 1 , 2 Mirkó Visontai 3 1 Eötvös University Budapest 2 Ecole Polytechnique Fédérale de Lausanne 3 University 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 Intersecting pairs of sets 1 Definitions History Previous results A new lower bound 2 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 Definitions A new lower bound History Summary Previous results Outline Intersecting pairs of sets 1 Definitions History Previous results A new lower bound 2 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 Definitions A new lower bound History Summary 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 Definitions A new lower bound History Summary 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 Definitions A new lower bound History Summary Previous results Intersecting pairs of sets Definition (Intersecting pairs of sets) An ( a , b ) -set system F is intersecting if for any ( A i , B i ) , ( A j , B j ) ∈ F with i � = j we have that A i ∩ B j � = ∅ and A j ∩ B i � = ∅ . Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets Definitions A new lower bound History Summary Previous results Intersecting pairs of sets Definition (Intersecting pairs of sets) An ( a , b ) -set system F is intersecting if for any ( A i , B i ) , ( A j , B j ) ∈ F with i � = j we have that A i ∩ B j � = ∅ and A j ∩ B i � = ∅ . Example F 1 = � ( { 1 , 2 } , { 3 , 4 , 5 } ) , ( { 1 , 3 } , { 4 , 5 , 6 } ) � is not intersecting, � � but F 2 = ( { 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 Definitions A new lower bound History Summary Previous results Outline Intersecting pairs of sets 1 Definitions History Previous results A new lower bound 2 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 Definitions A new lower bound History Summary Previous results Classical results Theorem (B. Bollobás ’65) The maximum possible size of an intersecting ( a , b ) -set system � a + b � is and this is sharp. b Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets Definitions A new lower bound History Summary Previous results Classical results Theorem (B. Bollobás ’65) The maximum possible size of an intersecting ( a , b ) -set system � a + b � is and this is sharp. b Theorem (P . Frankl ’82) If we require that A i ∩ B j � = ∅ for i > j only, then the same holds. � a + b � Namely, the maximum size of such a set system is . b Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets Definitions A new lower bound History Summary 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 ( A i , B i ) , ( A j , B j ) ∈ F with i � = j we have that A i ∩ B j and A j ∩ B i are not both empty. Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets Definitions A new lower bound History Summary 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 ( A i , B i ) , ( A j , B j ) ∈ F with i � = j we have that A i ∩ B j and A j ∩ B i 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 Definitions A new lower bound History Summary 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 Definitions A new lower bound History Summary Previous results The objective Definition Let g ( a , b ) denote the maximum possible size of a weakly intersecting system. � a + b � Problem: Investigate the ratio g ( a , b ) / . a Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets Definitions A new lower bound History Summary Previous results The objective Definition Let g ( a , b ) denote the maximum possible size of a weakly intersecting system. � a + b � Problem: Investigate the ratio g ( a , b ) / . a We will show that: � a + b � a + b →∞ g ( a , b ) / lim inf ≥ 2 − o ( 1 ) a . Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets Definitions A new lower bound History Summary Previous results Outline Intersecting pairs of sets 1 Definitions History Previous results A new lower bound 2 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 Definitions A new lower bound History Summary Previous results Lower bounds by Tuza Claim 1. g ( a , 1 ) ≥ 2 a + 1 . Zoltán Király, Zoltán L. Nagy, Dömötör Pálvölgyi, Mirkó Visontai On weakly intersecting pairs of sets

Intersecting pairs of sets Definitions A new lower bound History Summary Previous results Lower bounds by Tuza Claim 1. g ( a , 1 ) ≥ 2 a + 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 Definitions A new lower bound History Summary Previous results Lower bounds by Tuza Claim 1. g ( a , 1 ) ≥ 2 a + 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 Definitions A new lower bound History Summary 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 Definitions A new lower bound History Summary 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 Definitions A new lower bound History Summary 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