**Birkhoff’s Representation Theorem.** Every distributive lattice is isomorphic to a sublattice of the power set lattice of some set.

**Riesz–Markov–Kakutani Representation Theorem.** Any positive linear functional on the space of compactly supported continuous functions on a locally compact Hausdorff space can be viewed as integration against a measure.

**Kapovich-Millson Universality Theorem.** Any compact smooth manifold is diffeomorphic to a component of the configuration space of some planar linkage.

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What is ? What does it say about ? Try to calculate some values of and explain.

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Instead of looking at all possible drawings (which is infinite), we can partition the drawings into classes, depending on the rotation system of the drawing.

The following theorem guarantees us that the crossing number is still well-defined.

**Gioan’s Theorem.** Let D_{1} and D_{2} be two drawing of graph K_{n} in the plane. Then one can transform D_{1} into D_{2 }using only -moves. As a corollary, the crossing number of D_{1} and D_{2 }is the same.

Noted that in general this is not true when the graph is not complete.

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**Conjecture**[Thomas]**.**For any t, any sufficiently large t-connected graph with no -minor can be made planar by removing exactly t-5 vertices.

The case when t=6 has been proven. [Kawarabayashi-Norine-Thomas-Wollan ’12]- In a subgraph-closed graph family, having polynomial expansion is equivalent to having sublinear separator. [Dvořák-Norine ’15]
**Richter-Thomassen Conjecture**, now**Pach-Rubin-Tardos Theorem**, states that the total number of intersections between n pairwise intersecting Jordan curves in the plane, no three pass through the same point, is at least .

**Main Theorem**[Pach-Rubin-Tardos ’15]**.**Consider the above set of Jordan curves. Then the number of crossing points is times the number of touching points, those that are the only intersection between two Jordan curves.- Further results on arc and bar k-visibility graphs by Sawhney and Weed, 2016.
- Minimum cut of directed planar graphs in O(n log log n) time by Mozes, Nikolaev, Nussbaum, and Weimann, 2015.
- The weakly simple polygon result has been extended to geometric intersection numbers. [Despré-Lazarus ’15]

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Anyway, the following observation is really interesting; details can be found in the book “matching theory” by Lovasz and Plummer.

Consider a general graph . An equivalence relationship can be defined on the nodes of the graph as follows: two nodes and are *equivalent* if has no perfect matching(!).

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**Hindman’s theorem** (a special case). Suppose we color the natural numbers using k colors. Then there exists a color c and an infinite set S, all colored with c, such that every finite sum of over S has color c as well.

Also, this theorem is claimed to be *rather unpleasant to prove* if we insist on not using ultrafilters.

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