# Recent Activity

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

Author(s):

\begin{conjecture} Are all Mersenne Numbers with prime exponent ${2^p-1}$ Square free? \end{conjecture}

Keywords: Mersenne number

## What are hyperfuncoids isomorphic to? ★★

Author(s): Porton

Let $\mathfrak{A}$ be an indexed family of sets.

\emph{Products} are $\prod A$ for $A \in \prod \mathfrak{A}$.

\emph{Hyperfuncoids} are filters $\mathfrak{F} \Gamma$ on the lattice $\Gamma$ of all finite unions of products.

\begin{problem} Is $\bigcap^{\mathsf{\tmop{FCD}}}$ a bijection from hyperfuncoids $\mathfrak{F} \Gamma$ to: \begin{enumerate} \item prestaroids on $\mathfrak{A}$; \item staroids on $\mathfrak{A}$; \item completary staroids on $\mathfrak{A}$? \end{enumerate} If yes, is $\operatorname{up}^{\Gamma}$ defining the inverse bijection? If not, characterize the image of the function $\bigcap^{\mathsf{\tmop{FCD}}}$ defined on $\mathfrak{F} \Gamma$.

Consider also the variant of this problem with the set $\Gamma$ replaced with the set $\Gamma^{\ast}$ of complements of elements of the set $\Gamma$. \end{problem}

Keywords: hyperfuncoids; multidimensional

## Another conjecture about reloids and funcoids ★★

Author(s): Porton

\begin{definition} $\square f = \bigcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f$ for reloid $f$. \end{definition}

\begin{conjecture} $(\mathsf{RLD})_{\Gamma} f = \square (\mathsf{RLD})_{\mathrm{in}} f$ for every funcoid $f$. \end{conjecture}

Note: it is known that $(\mathsf{RLD})_{\Gamma} f \ne \square (\mathsf{RLD})_{\mathrm{out}} f$ (see below mentioned online article).

Keywords:

## Inequality for square summable complex series ★★

Author(s): Retkes

\begin{conjecture} For all $\alpha=(\alpha_1,\alpha_2,\ldots)\in l_2(\cal{C})$ the following inequality holds $$\sum_{n\geq 1}|\alpha_n|^2\geq \frac{6}{\pi^2}\sum_{k\geq0}\bigg| \sum_{l\geq0}\frac{1}{l+1}\alpha_{2^k(2l+1)}\bigg|^2 $$

\end{conjecture}

Keywords: Inequality

## One-way functions exist ★★★★

Author(s):

\begin{conjecture} \Def[One-way functions]{One-way_function} exist. \end{conjecture}

Keywords: one way function

## Graceful Tree Conjecture ★★★

Author(s):

\begin{conjecture} All trees are graceful \end{conjecture}

Keywords: combinatorics; graceful labeling

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

Author(s): Felsner; Hurtado; Noy; Streinu

Consider a set $S$ of great circles on a sphere with no three circles meeting at a point. The arrangement graph of $S$ 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.

\begin{conjecture} Every arrangement graph of a set of great circles is $3$-colourable. \end{conjecture}

Keywords: arrangement graph; graph coloring

## Chromatic Number of Common Graphs ★★

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

\begin{question} Do common graphs have bounded chromatic number? \end{question}

Keywords: common graph

## Erdős–Straus conjecture ★★

\begin{conjecture} % Enter your conjecture in LaTeX For all $n > 2$, there exist positive integers $x$, $y$, $z$ such that $$1/x + 1/y + 1/z = 4/n$$. % You may change "conjecture" to "question" or "problem" if you prefer. \end{conjecture}

Keywords: Egyptian fraction

## The 3n+1 conjecture ★★★

Author(s): Collatz

\begin{conjecture} Let $f(n) = 3n+1$ if $n$ is odd and $\frac{n}{2}$ if $n$ is even. Let $f(1) = 1$. Assume we start with some number $n$ and repeatedly take the $f$ of the current number. Prove that no matter what the initial number is we eventually reach $1$. \end{conjecture}

Keywords: integer sequence

## List Hadwiger Conjecture ★★

Author(s): Kawarabayashi; Mohar

\begin{conjecture} Every $K_t$-minor-free graph is $c t$-list-colourable for some constant $c\geq1$. \end{conjecture}

Keywords: Hadwiger conjecture; list colouring; minors

## Lucas Numbers Modulo m ★★

Author(s):

\begin{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. \end{conjecture}

Keywords: Lucas numbers

## Divisibility of central binomial coefficients ★★

Author(s): Graham

\begin{problem}[1] Prove that there exist infinitely many positive integers $n$ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7) = 1.$$ \end{problem}

\begin{problem}[2] Prove that there exists only a finite number of positive integers $n$ such that $$\gcd({2n\choose n}, 3\cdot 5\cdot 7\cdot 11) = 1.$$ \end{problem}

Keywords:

## ¿Are critical k-forests tight? ★★

Author(s): Strausz

\begin{conjecture} % Enter your conjecture in LaTeX Let $H$ be a $k$-\Def[uniform]{hypergraph} hypergraph. If $H$ is a critical $k$-forest, then it is a $k$-tree. \end{conjecture}

Keywords: heterochromatic number

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

Author(s): Morrison; Noel; Scott

\begin{question} Does there exist a constant $c>1/2$ and a function $n_0(k)$ such that if $|X|\geq n_0(k)$, then every saturated $k$-Sperner system $\mathcal{F}\subseteq \mathcal{P}(X)$ has cardinality at least $2^{(1+o(1))ck}$? \end{question}

Keywords: antichain; extremal combinatorics; minimum saturation; saturation; Sperner system

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

Author(s): Allagan

\begin{question} Given $a,b\geq2$, what is the smallest integer $t\geq0$ such that $\chi_\ell(K_{a,b}+K_t)= \chi(K_{a,b}+K_t)$? \end{question}

Keywords: complete bipartite graph; complete multipartite graph; list coloring

## Generalised Empty Hexagon Conjecture ★★

Author(s): Wood

\begin{conjecture} For each $\ell\geq3$ there is an integer $f(\ell)$ such that every set of at least $f(\ell)$ points in the plane contains $\ell$ collinear points or an empty hexagon. \end{conjecture}

Keywords: empty hexagon

## Nonrepetitive colourings of planar graphs ★★

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

\begin{question} Do planar graphs have bounded nonrepetitive chromatic number? \end{question}

Keywords: nonrepetitive colouring; planar graphs

## General position subsets ★★

Author(s): Gowers

\begin{question} What is the least integer $f(n)$ such that every set of at least $f(n)$ points in the plane contains $n$ collinear points or a subset of $n$ points in general position (no three collinear)? \end{question}

## Forcing a 2-regular minor ★★

\begin{conjecture} Every graph with average degree at least $\frac{4}{3}t-2$ contains every 2-regular graph on $t$ vertices as a minor. \end{conjecture}

Keywords: minors