# Recent Activity

## A funcoid related to directed topological spaces ★★

Author(s): Porton

**Conjecture**Let be the complete funcoid corresponding to the usual topology on extended real line . Let be the order on this set. Then is a complete funcoid.

**Proposition**It is easy to prove that is the infinitely small right neighborhood filter of point .

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

Keywords:

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

Author(s): Porton

**Conjecture**for principal funcoid and a set of funcoids of appropriate sources and destinations.

Keywords: distributivity; principal funcoid

## Weak saturation of the cube in the clique ★

**Problem**

Determine .

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 in the hypercube.

Keywords: cycles; extremal combinatorics; hypercube

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

**Problem**Determine the smallest percolating set for the -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 in the -dimensional hypercube?

Keywords: cycles; hypercube; minimum saturation; saturation

## Cycles in Graphs of Large Chromatic Number ★★

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

**Conjecture**If , then contains at least cycles of length .

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 containing no pair of orthogonal vectors is attained by two open caps of geodesic radius 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 -flow on is an orientation of together with a function from the edge set of into the real numbers such that , for all , and .

A -regular graph is a -graph if for every with odd.

**Conjecture**Let be an integer. If is a -graph, then .

Keywords: flow conjectures; nowhere-zero flows

## Circular flow number of regular class 1 graphs ★★

Author(s): Steffen

A nowhere-zero -flow on is an orientation of together with a function from the edge set of into the real numbers such that , for all , and . The circular flow number of is inf has a nowhere-zero -flow , and it is denoted by .

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

**Conjecture**Let be an integer and a -regular graph. If is a class 1 graph, then .

## Chromatic number of associahedron ★★

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

**Conjecture**Associahedra have unbounded chromatic number.

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

Author(s):

**Conjecture**A Mersenne prime is a Mersenne number that is prime.

Are there infinite number of Mersenne Primes?

Keywords: Mersenne number; Mersenne prime

## 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

## 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 ★★

**Conjecture**

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

Keywords: Egyptian fraction