# Recent Activity

## Are all Mersenne Numbers with prime exponent square-free? ★★★

Author(s):

Conjecture   Are all Mersenne Numbers with prime exponent Square free?

Keywords: Mersenne number

## What are hyperfuncoids isomorphic to? ★★

Author(s): Porton

Let be an indexed family of sets.

Products are for .

Hyperfuncoids are filters on the lattice of all finite unions of products.

Problem   Is a bijection from hyperfuncoids to:
\item prestaroids on ; \item staroids on ; \item completary staroids on ?

If yes, is defining the inverse bijection? If not, characterize the image of the function defined on .

Consider also the variant of this problem with the set replaced with the set of complements of elements of the set .

Keywords: hyperfuncoids; multidimensional

## Another conjecture about reloids and funcoids ★★

Author(s): Porton

Definition   for reloid .
Conjecture   for every funcoid .

Note: it is known that (see below mentioned online article).

Keywords:

## Inequality for square summable complex series ★★

Author(s): Retkes

Conjecture   For all the following inequality holds

Keywords: Inequality

## One-way functions exist ★★★★

Author(s):

Conjecture   One-way functions exist.

Keywords: one way function

## Graceful Tree Conjecture ★★★

Author(s):

Conjecture   All trees are graceful

Keywords: combinatorics; graceful labeling

## 3-Colourability of Arrangements of Great Circles ★★

Consider a set of great circles on a sphere with no three circles meeting at a point. The arrangement graph of has a vertex for each intersection point, and an edge for each arc directly connecting two intersection points. So this arrangement graph is 4-regular and planar.

Conjecture   Every arrangement graph of a set of great circles is -colourable.

Keywords: arrangement graph; graph coloring

## Chromatic Number of Common Graphs ★★

Author(s): Hatami; Hladký; Kráľ; Norine; Razborov

Question   Do common graphs have bounded chromatic number?

Keywords: common graph

## Erdős–Straus conjecture ★★

Author(s): Erdos; Straus

Conjecture

For all , there exist positive integers , , such that .

Keywords: Egyptian fraction

## The 3n+1 conjecture ★★★

Author(s): Collatz

Conjecture   Let if is odd and if is even. Let . Assume we start with some number and repeatedly take the of the current number. Prove that no matter what the initial number is we eventually reach .

Keywords: integer sequence

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

## 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