Improved Ramsey-type results in comparability graphs D aniel Kor - PowerPoint PPT Presentation

Improved Ramsey-type results in comparability graphs D´ aniel Kor´ andi EPFL May 14, 2019 joint work with Istv´ an Tomon

Partially ordered sets

Partially ordered sets Partially ordered set A poset is a set X with a transitive, antisymmetric relation < .

Partially ordered sets Partially ordered set A poset is a set X with a transitive, antisymmetric relation < . ◮ t -chain: x 1 < x 2 < · · · < x t

Partially ordered sets Partially ordered set A poset is a set X with a transitive, antisymmetric relation < . ◮ t -chain: x 1 < x 2 < · · · < x t ◮ r -antichain: x 1 , . . . , x r such that x i � < x j for every i , j

Partially ordered sets Partially ordered set A poset is a set X with a transitive, antisymmetric relation < . ◮ t -chain: x 1 < x 2 < · · · < x t ◮ r -antichain: x 1 , . . . , x r such that x i � < x j for every i , j Fact Every N -element poset contains a chain of length t or an antichain of size N / t (for every t ).

Partially ordered sets Partially ordered set A poset is a set X with a transitive, antisymmetric relation < . ◮ t -chain: x 1 < x 2 < · · · < x t ◮ r -antichain: x 1 , . . . , x r such that x i � < x j for every i , j Fact Every N -element poset contains a chain of length t or an antichain of size N / t (for every t ). “Dilworth”: If X has no t -chain, then it can be partitioned into t antichains.

Convex sets in the plane

Convex sets in the plane

Convex sets in the plane Question Given n convex sets in the plane, how big is the largest disjoint or pairwise intersecting subfamily?

Convex sets in the plane Question Given n convex sets in the plane, how big is the largest disjoint or pairwise intersecting subfamily?

Convex sets in the plane Question Given n convex sets in the plane, how big is the largest disjoint or pairwise intersecting subfamily?

Convex sets in the plane Question Given n convex sets in the plane, how big is the largest disjoint or pairwise intersecting subfamily? Theorem (Larman-Matouˇ sek-Pach-T¨ or˝ ocsik, 1994) At least n 1 / 5 .

Convex sets in the plane A , B disjoint, B is “below” A A B

Convex sets in the plane A , B disjoint, B is “below” A B A

Convex sets in the plane A , B disjoint, B is “below” A A B

Convex sets in the plane A , B disjoint, B is “below” A A B

Convex sets in the plane A , B disjoint, B is “below” A A A B B

Convex sets in the plane A , B disjoint, B is “below” A A A B B A B

Convex sets in the plane A , B disjoint, B is “below” A A A B B A A B B

Convex sets in the plane A , B disjoint, B is “below” A A < 1 B A < 2 B A A B B A A B B A < 3 B A < 4 B

Convex sets in the plane A , B disjoint, B is “below” A A < 1 B A < 2 B A A B B A A B B A < 3 B A < 4 B < 1 , < 2 , < 3 , < 4 are partial orders

Convex sets in the plane A , B disjoint, B is “below” A A < 1 B A < 2 B A A B B A A B B A < 3 B A < 4 B < 1 , < 2 , < 3 , < 4 are partial orders Sets A , B are disjoint iff they are comparable in any of < 1 , . . . , < 4 .

n convex sets in the plane Sets A , B are disjoint iff they are comparable in any of < 1 , . . . , < 4 .

n convex sets in the plane Sets A , B are disjoint iff they are comparable in any of < 1 , . . . , < 4 . Fact Every N -element poset contains a t -chain or an ( N / t )-antichain.

n convex sets in the plane Sets A , B are disjoint iff they are comparable in any of < 1 , . . . , < 4 . Fact Every N -element poset contains a t -chain or an ( N / t )-antichain. n 1 / 5 -chain < 1

n convex sets in the plane Sets A , B are disjoint iff they are comparable in any of < 1 , . . . , < 4 . Fact Every N -element poset contains a t -chain or an ( N / t )-antichain. n 1 / 5 -chain → disjoint family < 1

n convex sets in the plane Sets A , B are disjoint iff they are comparable in any of < 1 , . . . , < 4 . Fact Every N -element poset contains a t -chain or an ( N / t )-antichain. n 1 / 5 -chain → disjoint family < 1 Otherwise: n 4 / 5 -antichain S 1 .

n convex sets in the plane Sets A , B are disjoint iff they are comparable in any of < 1 , . . . , < 4 . Fact Every N -element poset contains a t -chain or an ( N / t )-antichain. n 1 / 5 -chain → disjoint family < 1 Otherwise: n 4 / 5 -antichain S 1 . n 1 / 5 -chain in S 1 < 2

n convex sets in the plane Sets A , B are disjoint iff they are comparable in any of < 1 , . . . , < 4 . Fact Every N -element poset contains a t -chain or an ( N / t )-antichain. n 1 / 5 -chain → disjoint family < 1 Otherwise: n 4 / 5 -antichain S 1 . n 1 / 5 -chain in S 1 → disjoint family < 2

n convex sets in the plane Sets A , B are disjoint iff they are comparable in any of < 1 , . . . , < 4 . Fact Every N -element poset contains a t -chain or an ( N / t )-antichain. n 1 / 5 -chain → disjoint family < 1 Otherwise: n 4 / 5 -antichain S 1 . n 1 / 5 -chain in S 1 → disjoint family < 2 Otherwise: n 3 / 5 -antichain S 2 ⊆ S 1 .

n convex sets in the plane Sets A , B are disjoint iff they are comparable in any of < 1 , . . . , < 4 . Fact Every N -element poset contains a t -chain or an ( N / t )-antichain. n 1 / 5 -chain → disjoint family < 1 Otherwise: n 4 / 5 -antichain S 1 . n 1 / 5 -chain in S 1 → disjoint family < 2 Otherwise: n 3 / 5 -antichain S 2 ⊆ S 1 . n 1 / 5 -chain in S 2 → disjoint family < 3 Otherwise: n 2 / 5 -antichain S 3 ⊆ S 2 .

n convex sets in the plane Sets A , B are disjoint iff they are comparable in any of < 1 , . . . , < 4 . Fact Every N -element poset contains a t -chain or an ( N / t )-antichain. n 1 / 5 -chain → disjoint family < 1 Otherwise: n 4 / 5 -antichain S 1 . n 1 / 5 -chain in S 1 → disjoint family < 2 Otherwise: n 3 / 5 -antichain S 2 ⊆ S 1 . n 1 / 5 -chain in S 2 → disjoint family < 3 Otherwise: n 2 / 5 -antichain S 3 ⊆ S 2 . n 1 / 5 -chain in S 3 → disjoint family < 4 Otherwise: n 1 / 5 -antichain S 4 ⊆ S 3 .

n convex sets in the plane Sets A , B are disjoint iff they are comparable in any of < 1 , . . . , < 4 . Fact Every N -element poset contains a t -chain or an ( N / t )-antichain. n 1 / 5 -chain → disjoint family < 1 Otherwise: n 4 / 5 -antichain S 1 . n 1 / 5 -chain in S 1 → disjoint family < 2 Otherwise: n 3 / 5 -antichain S 2 ⊆ S 1 . n 1 / 5 -chain in S 2 → disjoint family < 3 Otherwise: n 2 / 5 -antichain S 3 ⊆ S 2 . n 1 / 5 -chain in S 3 → disjoint family < 4 Otherwise: n 1 / 5 -antichain S 4 ⊆ S 3 . ⇒ S 4 incomparable in all 4 posets → intersecting family.

Graph language

Graph language Comparability graph of a poset Connect a , b with an edge if a < b or b < a .

Graph language Comparability graph of a poset Connect a , b with an edge if a < b or b < a . Lemma If G is the union of k comparability graphs, then G contains a 1 k +1 . clique or independent set of size n

Graph language Comparability graph of a poset Connect a , b with an edge if a < b or b < a . Lemma If G is the union of k comparability graphs, then G contains a 1 k +1 . clique or independent set of size n n 1 / 5 convex sets

Graph language Comparability graph of a poset Connect a , b with an edge if a < b or b < a . Lemma If G is the union of k comparability graphs, then G contains a 1 k +1 . clique or independent set of size n n 1 / 5 convex sets n 1 / 3 halflines

Graph language Comparability graph of a poset Connect a , b with an edge if a < b or b < a . Lemma If G is the union of k comparability graphs, then G contains a 1 k +1 . clique or independent set of size n n 1 / 5 convex sets n 1 / 3 halflines

Graph language Comparability graph of a poset Connect a , b with an edge if a < b or b < a . Lemma If G is the union of k comparability graphs, then G contains a 1 k +1 . clique or independent set of size n n 1 / 5 convex sets n 1 / 3 halflines

Graph language Comparability graph of a poset Connect a , b with an edge if a < b or b < a . Lemma If G is the union of k comparability graphs, then G contains a 1 k +1 . clique or independent set of size n n 1 / 5 convex sets n 1 / 3 halflines

Graph language Comparability graph of a poset Connect a , b with an edge if a < b or b < a . Lemma If G is the union of k comparability graphs, then G contains a 1 k +1 . clique or independent set of size n n 1 / 5 convex sets n 1 / 3 halflines

Graph language Comparability graph of a poset Connect a , b with an edge if a < b or b < a . Lemma If G is the union of k comparability graphs, then G contains a 1 k +1 . clique or independent set of size n Lower bound n 1 / 5 convex sets n 1 / 3 halflines

Graph language Comparability graph of a poset Connect a , b with an edge if a < b or b < a . Lemma If G is the union of k comparability graphs, then G contains a 1 k +1 . clique or independent set of size n Lower bound Upper bound n 0 . 405 (Kynˇ n 1 / 5 convex sets cl) n 0 . 431 (LMPT) n 1 / 3 halflines