
Recent Activity
Hedetniemi's Conjecture ★★★
Author(s): Hedetniemi


Here is the tensor product (also called the direct or categorical product) of
and
.
Keywords: categorical product; coloring; homomorphism; tensor product
Diophantine quintuple conjecture ★★
Author(s):




It would follow from the following stronger conjecture [Da]:



Keywords:
Several ways to apply a (multivalued) multiargument function to a family of filters ★★★
Author(s): Porton

1. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the reloidal product of filters .
2. The funcoid corresponding to this function (considered as a single argument function on indexed families) applied to the starred reloidal product of filters .
3. .
Keywords: funcoid; function; multifuncoid; staroid
Jones' conjecture ★★
For a graph , let
denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let
denote the cardinality of a minimum feedback vertex set (set of vertices
so that
is acyclic).


Keywords: cycle packing; feedback vertex set; planar graph
Multicolour Erdős--Hajnal Conjecture ★★★










Keywords: ramsey theory
Sidorenko's Conjecture ★★★
Author(s): Sidorenko





Keywords: density problems; extremal combinatorics; homomorphism
Edge-Unfolding Convex Polyhedra ★★
Author(s): Shephard
Point sets with no empty pentagon ★
Author(s): Wood
Keywords: combinatorial geometry; visibility graph
Singmaster's conjecture ★★
Author(s): Singmaster

The number appears once in Pascal's triangle,
appears twice,
appears three times, and
appears
times. There are infinite families of numbers known to appear
times. The only number known to appear
times is
. It is not known whether any number appears more than
times. The conjectured upper bound could be
; Singmaster thought it might be
or
. See Singmaster's conjecture.
Keywords: Pascal's triangle
Waring rank of determinant ★★
Author(s): Teitler

For simplicity say we work over the complex numbers. The generic matrix is the matrix with entries
for
. Its determinant is a homogeneous form of degree
, in
variables. If
is a homogeneous form of degree
, a power sum expression for
is an expression of the form
, the
(homogeneous) linear forms. The Waring rank of
is the least number of terms
in any power sum expression for
. For example, the expression
means that
has Waring rank
(it can't be less than
, as
).
The generic determinant
(or
) has Waring rank
. The Waring rank of the
generic determinant is at least
and no more than
, see for instance Lower bound for ranks of invariant forms, Example 4.1. The Waring rank of the permanent is also of interest. The comparison between the determinant and permanent is potentially relevant to Valiant's "VP versus VNP" problem.
Keywords: Waring rank, determinant
Monochromatic vertex colorings inherited from Perfect Matchings ★★★
Author(s):





Keywords:
Cycle Double Covers Containing Predefined 2-Regular Subgraphs ★★★
Author(s): Arthur; Hoffmann-Ostenhof








Keywords:
Monochromatic reachability in arc-colored digraphs ★★★
Author(s): Sands; Sauer; Woodrow








Keywords:
3-Decomposition Conjecture ★★★
Author(s): Arthur; Hoffmann-Ostenhof

Keywords: cubic graph
Which outer reloids are equal to inner ones ★★
Author(s): Porton
Warning: This formulation is vague (not exact).

The problem seems rather difficult.
Keywords:
A diagram about funcoids and reloids ★★
Author(s): Porton
Define for posets with order :
;
.
Note that the above is a generalization of monotone Galois connections (with and
replaced with suprema and infima).
Then we have the following diagram:
What is at the node "other" in the diagram is unknown.





Keywords: Galois connections
Outward reloid of composition vs composition of outward reloids ★★
Author(s): Porton



Keywords: outward reloid
Sum of prime and semiprime conjecture ★★
Author(s): Geoffrey Marnell

A funcoid related to directed topological spaces ★★
Author(s): Porton

![$ [-\infty,+\infty] = \mathbb{R}\cup\{-\infty,+\infty\} $](/files/tex/3252019c60a83f00ff396d823dbff8040639f409.png)



![$ x\in[-\infty,+\infty] $](/files/tex/4e57a21194d8d5a659e259a111ed13a9c23b52a1.png)
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



Keywords: distributivity; principal funcoid