# Recent Activity

## List Colourings of Complete Multipartite Graphs with 2 Big Parts ★★

Author(s): Allagan

Question   Given , what is the smallest integer such that ?

## Generalised Empty Hexagon Conjecture ★★

Author(s): Wood

Conjecture   For each there is an integer such that every set of at least points in the plane contains collinear points or an empty hexagon.

Keywords: empty hexagon

## Nonrepetitive colourings of planar graphs ★★

Author(s): Alon N.; Grytczuk J.; Hałuszczak M.; Riordan O.

Question   Do planar graphs have bounded nonrepetitive chromatic number?

Keywords: nonrepetitive colouring; planar graphs

## General position subsets ★★

Author(s): Gowers

Question   What is the least integer such that every set of at least points in the plane contains collinear points or a subset of points in general position (no three collinear)?

## Forcing a 2-regular minor ★★

Author(s): Reed; Wood

Conjecture   Every graph with average degree at least contains every 2-regular graph on vertices as a minor.

Keywords: minors

Author(s): Harvey; Reed; Seymour; Wood

Conjecture   For every graph ,
(a)
(b)
(c) .

Keywords: fractional coloring, minors

## Generalized path-connectedness in proximity spaces ★★

Author(s): Porton

Let be a proximity.

A set is connected regarding iff .

Conjecture   The following statements are equivalent for every endofuncoid and a set :
\item is connected regarding . \item For every there exists a totally ordered set such that , , and for every partion of into two sets , such that , we have .

Keywords: connected; connectedness; proximity space

## Direct proof of a theorem about compact funcoids ★★

Author(s): Porton

Conjecture   Let is a -separable (the same as for symmetric transitive) compact funcoid and is a uniform space (reflexive, symmetric, and transitive endoreloid) such that . Then .

The main purpose here is to find a direct proof of this conjecture. It seems that this conjecture can be derived from the well known theorem about existence of exactly one uniformity on a compact set. But that would be what I call an indirect proof, we need a direct proof instead.

The direct proof may be constructed by correcting all errors an omissions in this draft article.

Direct proof could be better because with it we would get a little more general statement like this:

Conjecture   Let be a -separable compact reflexive symmetric funcoid and be a reloid such that
\item ; \item .

Then .

## Dirac's Conjecture ★★

Author(s): Dirac

Conjecture   For every set of points in the plane, not all collinear, there is a point in contained in at least lines determined by , for some constant .

Keywords: point set

## Separators in string graphs ★★

Author(s): Fox; Pach; Tóth

Conjecture   Every string graph with edges has a separator of size .

Keywords: separator; string graphs

## Roller Coaster permutations ★★★

Author(s): Ahmed; Snevily

Let denote the set of all permutations of . Let and denote respectively the number of increasing and the number of decreasing sequences of contiguous numbers in . Let denote the set of subsequences of with length at least three. Let denote .

A permutation is called a Roller Coaster permutation if . Let be the set of all Roller Coaster permutations in .

Conjecture   For ,
\item If , then . \item If , then with .
Conjecture  (Odd Sum conjecture)   Given ,
\item If , then is odd for . \item If , then for all .

Keywords:

## Rota's basis conjecture ★★★

Author(s): Rota

Conjecture   Let be a vector space of dimension and let be bases. Then there exist disjoint transversals of each of which is a base.

Keywords: base; latin square; linear algebra; matroid; transversal

## Graphs of exact colorings ★★

Author(s):

Conjecture For , let be the statement that given any exact -coloring of the edges of a complete countably infinite graph (that is, a coloring with colors all of which must be used at least once), there exists an exactly -colored countably infinite complete subgraph. Then is true if and only if , , or .

Keywords:

## Imbalance conjecture ★★

Author(s): Kozerenko

Conjecture   Suppose that for all edges we have . Then is graphic.

Keywords: edge imbalance; graphic sequences

## Every metamonovalued reloid is monovalued ★★

Author(s): Porton

Conjecture   Every metamonovalued reloid is monovalued.

Keywords:

## Every metamonovalued funcoid is monovalued ★★

Author(s): Porton

Conjecture   Every metamonovalued funcoid is monovalued.

The reverse is almost trivial: Every monovalued funcoid is metamonovalued.

Keywords: monovalued

## Decomposition of completions of reloids ★★

Author(s): Porton

Conjecture   For composable reloids and it holds
\item if is a co-complete reloid; \item if is a complete reloid; \item ; \item ; \item .

Keywords: co-completion; completion; reloid

## List Total Colouring Conjecture ★★

Author(s): Borodin; Kostochka; Woodall

Conjecture   If is the total graph of a multigraph, then .

Keywords: list coloring; Total coloring; total graphs

## Partitioning the Projective Plane ★★

Author(s): Noel

Throughout this post, by projective plane we mean the set of all lines through the origin in .

Definition   Say that a subset of the projective plane is octahedral if all lines in pass through the closure of two opposite faces of a regular octahedron centered at the origin.
Definition   Say that a subset of the projective plane is weakly octahedral if every set such that is octahedral.
Conjecture   Suppose that the projective plane can be partitioned into four sets, say and such that each set is weakly octahedral. Then each is octahedral.

Keywords: Partitioning; projective plane

## Kriesell's Conjecture ★★

Author(s): Kriesell

Conjecture   Let be a graph and let such that for any pair there are edge-disjoint paths from to in . Then contains edge-disjoint trees, each of which contains .