Mathematical Coloring Book – Mike Pawliuk

Bootcamp 3 – Ramsey DocCourse Prague 2016

The following notes are from the Ramsey DocCourse in Prague 2016. The notes are taken by me and I have edited them. In the process I may have introduced some errors; email me or comment below and I will happily fix them.

Title: Bootcamp (3 of 6)

Lecturer: Jaroslav Nešetřil

Date: Friday September 23, 2016.

Main Topics: Point Ramsey for graphs, $\langle A,B,C \rangle$ hypergraphs.

Definitions: Ramsey property for finite structures, Ramsey Class, point-Ramsey, edge-Ramsey, $\langle A,B,C \rangle$ hypergraphs, Chromatic number, Ordering Property.

Bootcamp 1 – Bootcamp 2 – Bootcamp 3 – Bootcamp 4 – Bootcamp 5 – Bootcamp 6 – Bootcamp 7

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Helly’s Theorem (2/2)

Last week we looked at the concepts of a collection of sets being n-linked or having the finite intersection property. The key theorem was Helly’s theorem which says:

Helly’s Theorem: If a (countable) family of closed convex sets (at least one of which is bounded) in the plane are 3-linked, then they have a point in common, as they have the FIP.

Now I will look at some of the generalizations that Alexander Soifer, author of “The Mathematical Coloring Book”, makes in Chapter 28 of that book. More than pure generalizations they are the combination of Ramsey theory and Helly’s Theorem

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