# Recent Activity

## 3-flow conjecture ★★★

Author(s): Tutte

Conjecture   Every 4-edge-connected graph has a nowhere-zero 3-flow.

Keywords: nowhere-zero flow

## Invariant subspace problem ★★★

Author(s):

Problem   Does every bounded linear operator on an infinite-dimensional separable Hilbert space have a non-trivial closed invariant subspace?

Keywords: subspace

## Sets with distinct subset sums ★★★

Author(s): Erdos

Say that a set has distinct subset sums if distinct subsets of have distinct sums.

Conjecture   There exists a fixed constant so that whenever has distinct subset sums.

Keywords: subset sum

## Seymour's Second Neighbourhood Conjecture ★★★

Author(s): Seymour

Conjecture   Any oriented graph has a vertex whose outdegree is at most its second outdegree.

Keywords: Caccetta-Häggkvist; neighbourhood; second; Seymour

## Which lattices occur as intervals in subgroup lattices of finite groups? ★★★★

Author(s):

Conjecture

There exists a finite lattice that is not an interval in the subgroup lattice of a finite group.

Keywords: congruence lattice; finite groups

## Quartic rationally derived polynomials ★★★

Author(s): Buchholz; MacDougall

Call a polynomial rationally derived if all roots of and the nonzero derivatives of are rational.

Conjecture   There does not exist a quartic rationally derived polynomial with four distinct roots.

Keywords: derivative; diophantine; elliptic; polynomial

## Nonseparating planar continuum ★★

Author(s):

Conjecture   Does any path-connected, compact set in the plane which does not separate the plane have the fixed point property?

A set has the fixed point property if every continuous map from it into itself has a fixed point.

Keywords: fixed point

## Hilbert-Smith conjecture ★★

Author(s): David Hilbert; Paul A. Smith

Conjecture   Let be a locally compact topological group. If has a continuous faithful group action on an -manifold, then is a Lie group.

Keywords:

## trace inequality ★★

Author(s):

Let be positive semidefinite, by Jensen's inequality, it is easy to see , whenever .

What about the , is it still valid?

Keywords:

## Real roots of the flow polynomial ★★

Author(s): Welsh

Conjecture   All real roots of nonzero flow polynomials are at most 4.

Keywords: flow polynomial; nowhere-zero flow

## Hamiltonicity of Cayley graphs ★★★

Author(s): Rapaport-Strasser

Question   Is every Cayley graph Hamiltonian?

Keywords:

## Finite Lattice Representation Problem ★★★★

Author(s):

Conjecture

There exists a finite lattice which is not the congruence lattice of a finite algebra.

Keywords: congruence lattice; finite algebra

## Outer reloid of restricted funcoid ★★

Author(s): Porton

Question   for every filter objects and and a funcoid ?

Keywords: direct product of filters; outer reloid

## Star chromatic index of complete graphs ★★

Author(s): Dvorak; Mohar; Samal

Conjecture   Is it possible to color edges of the complete graph using colors, so that the coloring is proper and no 4-cycle and no 4-edge path is using only two colors?

Equivalently: is the star chromatic index of linear in ?

Keywords: complete graph; edge coloring; star coloring

## Star chromatic index of cubic graphs ★★

Author(s): Dvorak; Mohar; Samal

The star chromatic index of a graph is the minimum number of colors needed to properly color the edges of the graph so that no path or cycle of length four is bi-colored.

Question   Is it true that for every (sub)cubic graph , we have ?

Keywords: edge coloring; star coloring

## Inscribed Square Problem ★★

Author(s): Toeplitz

Conjecture   Does every Jordan curve have 4 points on it which form the vertices of a square?

Keywords: simple closed curve; square

## Lindelöf hypothesis ★★

Author(s): Lindelöf

Conjecture   For any

Since can be replaced by a smaller value, we can also write the conjecture as, for any positive ,

Keywords: Riemann Hypothesis; zeta

## Termination of the sixth Goodstein Sequence ★

Author(s): Graham

Question   How many steps does it take the sixth Goodstein sequence to terminate?

Keywords: Goodstein Sequence

## Consecutive non-orientable embedding obstructions ★★★

Author(s):

Conjecture   Is there a graph that is a minor-minimal obstruction for two non-orientable surfaces?

Keywords: minor; surface

## Diagonal Ramsey numbers ★★★★

Author(s): Erdos

Let denote the diagonal Ramsey number.

Conjecture   exists.
Problem   Determine the limit in the above conjecture (assuming it exists).

Keywords: Ramsey number