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A funcoid related to directed topological spaces ★★

Author(s): Porton

Conjecture   Let $ R $ be the complete funcoid corresponding to the usual topology on extended real line $ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $. Let $ \geq $ be the order on this set. Then $ R\sqcap^{\mathsf{FCD}}\mathord{\geq} $ is a complete funcoid.
Proposition   It is easy to prove that $ \langle R\sqcap^{\mathsf{FCD}}\mathord{\geq}\rangle \{x\} $ is the infinitely small right neighborhood filter of point $ x\in[-\infty,+\infty] $.

If proved true, the conjecture then can be generalized to a wider class of posets.


Infinite distributivity of meet over join for a principal funcoid ★★

Author(s): Porton

Conjecture   $ f \sqcap \bigsqcup S = \bigsqcup \langle f \sqcap \rangle^{\ast} S $ for principal funcoid $ f $ and a set $ S $ of funcoids of appropriate sources and destinations.

Keywords: distributivity; principal funcoid

Weak saturation of the cube in the clique

Author(s): Morrison; Noel


Determine $ \text{wsat}(K_n,Q_3) $.

Keywords: bootstrap percolation; hypercube; Weak saturation

Convex Equipartitions with Extreme Perimeter ★★

Author(s): Nandakumar

To divide a given 2D convex region C into a specified number n of convex pieces all of equal area (perimeters could be different) such that the total perimeter of pieces is (1) maximized (2) minimized.

Remark: It appears maximizing the total perimeter is the easier problem.

Keywords: convex equipartition

Turán Problem for $10$-Cycles in the Hypercube ★★

Author(s): Erdos

Problem   Bound the extremal number of $ C_{10} $ in the hypercube.

Keywords: cycles; extremal combinatorics; hypercube

Extremal $4$-Neighbour Bootstrap Percolation in the Hypercube ★★

Author(s): Morrison; Noel

Problem   Determine the smallest percolating set for the $ 4 $-neighbour bootstrap process in the hypercube.

Keywords: bootstrap percolation; extremal combinatorics; hypercube; percolation

Saturation in the Hypercube ★★

Author(s): Morrison; Noel; Scott

Question   What is the saturation number of cycles of length $ 2\ell $ in the $ d $-dimensional hypercube?

Keywords: cycles; hypercube; minimum saturation; saturation

Cycles in Graphs of Large Chromatic Number ★★

Author(s): Brewster; McGuinness; Moore; Noel

Conjecture   If $ \chi(G)>k $, then $ G $ contains at least $ \frac{(k+1)(k-1)!}{2} $ cycles of length $ 0\bmod k $.

Keywords: chromatic number; cycles

The Double Cap Conjecture ★★

Author(s): Kalai

Conjecture   The largest measure of a Lebesgue measurable subset of the unit sphere of $ \mathbb{R}^n $ containing no pair of orthogonal vectors is attained by two open caps of geodesic radius $ \pi/4 $ around the north and south poles.

Keywords: combinatorial geometry; independent set; orthogonality; projective plane; sphere

Circular flow numbers of $r$-graphs ★★

Author(s): Steffen

A nowhere-zero $ r $-flow $ (D(G),\phi) $ on $ G $ is an orientation $ D $ of $ G $ together with a function $ \phi $ from the edge set of $ G $ into the real numbers such that $ 1 \leq |\phi(e)| \leq r-1 $, for all $ e \in E(G) $, and $ \sum_{e \in E^+(v)}\phi(e) = \sum_{e \in E^-(v)}\phi(e), \textrm{ for all } v \in V(G) $.

A $ (2t+1) $-regular graph $ G $ is a $ (2t+1) $-graph if $ |\partial_G(X)| \geq 2t+1 $ for every $ X \subseteq V(G) $ with $ |X| $ odd.

Conjecture   Let $ t > 1 $ be an integer. If $ G $ is a $ (2t+1) $-graph, then $ F_c(G) \leq 2 + \frac{2}{t} $.

Keywords: flow conjectures; nowhere-zero flows

Circular flow number of regular class 1 graphs ★★

Author(s): Steffen

A nowhere-zero $ r $-flow $ (D(G),\phi) $ on $ G $ is an orientation $ D $ of $ G $ together with a function $ \phi $ from the edge set of $ G $ into the real numbers such that $ 1 \leq |\phi(e)| \leq r-1 $, for all $ e \in E(G) $, and $ \sum_{e \in E^+(v)}\phi(e) = \sum_{e \in E^-(v)}\phi(e), \textrm{ for all } v \in V(G) $. The circular flow number of $ G $ is inf$ \{ r | G $ has a nowhere-zero $ r $-flow $ \} $, and it is denoted by $ F_c(G) $.

A graph with maximum vertex degree $ k $ is a class 1 graph if its edge chromatic number is $ k $.

Conjecture   Let $ t \geq 1 $ be an integer and $ G $ a $ (2t+1) $-regular graph. If $ G $ is a class 1 graph, then $ F_c(G) \leq 2 + \frac{2}{t} $.

Keywords: nowhere-zero flow, edge-colorings, regular graphs

Chromatic number of associahedron ★★

Author(s): Fabila-Monroy; Flores-Penaloza; Huemer; Hurtado; Urrutia; Wood

Conjecture   Associahedra have unbounded chromatic number.

Keywords: associahedron, graph colouring, chromatic number

Are there infinite number of Mersenne Primes? ★★★★


Conjecture   A Mersenne prime is a Mersenne number \[ M_n  = 2^p  - 1 \] that is prime.

Are there infinite number of Mersenne Primes?

Keywords: Mersenne number; Mersenne prime

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


Conjecture   Are all Mersenne Numbers with prime exponent $ {2^p-1} $ Square free?

Keywords: Mersenne number

What are hyperfuncoids isomorphic to? ★★

Author(s): Porton

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

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

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

Problem   Is $ \bigcap^{\mathsf{\tmop{FCD}}} $ a bijection from hyperfuncoids $ \mathfrak{F} \Gamma $ to:
    \item prestaroids on $ \mathfrak{A} $; \item staroids on $ \mathfrak{A} $; \item completary staroids on $ \mathfrak{A} $?

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 $.

Keywords: hyperfuncoids; multidimensional

Another conjecture about reloids and funcoids ★★

Author(s): Porton

Definition   $ \square f = \bigcap^{\mathsf{RLD}} \mathrm{up}^{\Gamma (\operatorname{Src} f ; \operatorname{Dst} f)} f $ for reloid $ f $.
Conjecture   $ (\mathsf{RLD})_{\Gamma} f = \square (\mathsf{RLD})_{\mathrm{in}} f $ for every funcoid $ f $.

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


Inequality for square summable complex series ★★

Author(s): Retkes

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

Keywords: Inequality

One-way functions exist ★★★★


Conjecture   One-way functions exist.

Keywords: one way function

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


For all $ n > 2 $, there exist positive integers $ x $, $ y $, $ z $ such that $$1/x + 1/y + 1/z = 4/n$$.

Keywords: Egyptian fraction