# Recent Activity

## List Hadwiger Conjecture ★★

Author(s): Kawarabayashi; Mohar

Conjecture   Every -minor-free graph is -list-colourable for some constant .

Keywords: Hadwiger conjecture; list colouring; minors

## Lucas Numbers Modulo m ★★

Author(s):

Conjecture   The sequence {L(n) mod m}, where L(n) are the Lucas numbers, contains a complete residue system modulo m if and only if m is one of the following: 2, 4, 6, 7, 14, 3^k, k >=1.

Keywords: Lucas numbers

## Divisibility of central binomial coefficients ★★

Author(s): Graham

Problem  (1)   Prove that there exist infinitely many positive integers such that
Problem  (2)   Prove that there exists only a finite number of positive integers such that

Keywords:

## ¿Are critical k-forests tight? ★★

Author(s): Strausz

Conjecture

Let be a -uniform hypergraph. If is a critical -forest, then it is a -tree.

Keywords: heterochromatic number

## Saturated $k$-Sperner Systems of Minimum Size ★★

Author(s): Morrison; Noel; Scott

Question   Does there exist a constant and a function such that if , then every saturated -Sperner system has cardinality at least ?

## 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

## 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

## Fractional Hadwiger ★★

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

## 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