**18**

votes

**5**answers

947 views

### Collection of conjectures and open problems in graph theory

Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?

**5**

votes

**1**answer

355 views

### Ruth-Aaron triples, etc

A Ruth-Aaron pair is two numbers $(n,n+1)$ such that
their sum of prime factors is equal, counting repeated prime factors.
(The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!)
So
...

**10**

votes

**0**answers

277 views

### Is every projective space curve a set-theoretic intersection of two surfaces? What is known about this question?

I am sorry if this question has been already asked, couldn't find anything similar myself. I have recently recalled this long standing open problem of whether every irreducible curve in ...

**2**

votes

**0**answers

62 views

### Long paths in the supercritical percolation.

I have a question on the length of the longest path, denoted by $\ell_n$, in the supercritical percolation on $[0,n]^d$, denoted by $C_n$.
We know that $C_n$ has a giant component whose size is of ...

**1**

vote

**1**answer

307 views

### Improving the bound $q < n\sqrt{3}$ for an odd perfect number $N = {q^k}{n^2}$ given in Eulerian form

(Note: This was cross-posted from MSE.)
Let $N = {q^k}{n^2}$ be an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q, n) = 1$).
Therefore, ...

**4**

votes

**0**answers

120 views

### Reduction argument from a general vertex set V(G) to a prime power in Prof. Keevash's proof on the Existence of Designs

The proof flow of the paper "On the Existence of Designs" by Prof. Keevash as I understand it is the following:
-- Reduction from the general case to $V = \mathbb{F}_{p^a}$ (Lemma 6.3)
-- Covering ...

**6**

votes

**1**answer

212 views

### Inverted pair of complex analytic families

I read the following "problem" in an old set of notes of Morrow and Kodaira which focused on deformations of complex manifolds:
Find a pair of complex analytic families $\lbrace M_t\rbrace$ and ...

**8**

votes

**0**answers

126 views

### Randomly placing nonoverlapping unit cuboids

Suppose one places unit cuboids of dimension $d$ with min-corners
uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden.
At some point, the region is "saturated," ...

**2**

votes

**2**answers

240 views

### Gauss Codes that produce classical knots as opposed to virtual knots

I have been doing some research in Gauss codes and have been reading Kauffman's paper Virtual Knot Theory. In section 3.3, Theorem 2, he states that
If $K$ is a virtual knot whose underlying Gauss ...

**13**

votes

**0**answers

309 views

### Does $S^4$ have a “symplecto-homeomorphic” structure?

The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms ...

**27**

votes

**3**answers

830 views

### Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point.
Theorem. Suppose $P$ and $Q$ are posets ...

**3**

votes

**1**answer

86 views

### reduction to np hard ordering problem

I am trying to show a reduction from a problem of ordering problem to an np-hard problem that has approximation poly-time algorithm.
My problem is: I have M auctions and in each auction I have N ...

**1**

vote

**1**answer

70 views

### A possible minimal aperiodic set of corner Wang Tile

From one of my previous question Aperiodic set of corner Wang Tile (although it is put on hold), I realize there is a systematic way to construct aperiodic corner type of Wang tile from edge type ...

**2**

votes

**0**answers

152 views

### Both NP-hard but different [closed]

What's the fundamental difference between the Knapsack problem and the travelling salesman (TSP) problem both of which are NP-hard, while the reality is that TSP could be solved much much faster?

**6**

votes

**2**answers

283 views

### Update on list of open problems for Cherednik/Symplectic Reflection Algebras

Background:
There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...

**6**

votes

**3**answers

512 views

### Is the Manickam-Miklós-Singhi Conjecture solved?

This arXiv paper is claimed to contain a proof for the MMS conjecture. But it seems that this manuscript is not yet peer reviewed by other mathematicians. I personally tried to follow the paper, but ...

**9**

votes

**1**answer

236 views

### Subspaces of $l_{1}$ are not Lipschitz complemented in $l_{1}$

I have thought about the following question for several years. This question may be stupid or not interesting. My question is: Is there a subspace $U$ of $l{_1}$ such that the quotient $l_{1}/U$ is ...

**10**

votes

**1**answer

242 views

### Is there a non-zero ghost map between finite suspension spectra?

A morphism $f\colon X\to Y$ of spectra such that for every integer $n$ the induced map $\pi_n(f)\colon\pi_n(X)\to\pi_n(Y)$ on stable homotopy groups is zero is called a ghost map.
Not every ghost map ...

**10**

votes

**1**answer

255 views

### A special case of the integer Hodge conjecture

Let $X$ be a projective complex manifold of dimension $n$.
Are torsion cohomology classes in $H^{2n-2}(X,\mathbb{Z})$ algebraic?
(We may assume, without loss of generality, that $n=3$, because of the ...

**4**

votes

**0**answers

176 views

### Packing space by cones: Translates best?

Let $C$ be a right circular cone, the convex hull of a unit-radius disk
and a point directly above the disk center at height $h$.
Is the ...

**0**

votes

**0**answers

154 views

### Is there a “natural” reformulation of Hodge conjecture in terms of L-functions?

I just glanced at the Wikipedia article about the Hodge conjecture, and a (probably very naive, due to my huge lack of knowledge of the subject) question just came to my mind: can one associate ...

**11**

votes

**0**answers

278 views

### Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$.
If there are $g(n)$ non-isomorphic groups of order $n$,
ideally each group would occur with probability ...

**1**

vote

**1**answer

90 views

### simple cycle analog in 2D (with application in tiling)

We know that any closed cycle of a graph could be decomposed into sum of simple cycles. To translate this theorem into tiling of 1D (Wang tile). We know that any 1D periodic tiling could be ...

**7**

votes

**1**answer

397 views

### Is there a precise notion of “almost all” such that almost all finite groups are Galois groups of extensions of the rationals?

I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to ...

**1**

vote

**1**answer

174 views

### relationship between corner tile and edge tile of wang tile

It is clear that any corner type of Wang Tile could be converted to edge type of Wang Tile by defining the edge color according to the corner color.
However, could we convert edge type of Wang Tile ...

**36**

votes

**1**answer

1k views

### improving known bounds for Pierce expansions; cash prize

Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have ...

**6**

votes

**1**answer

164 views

### Integral straight-line embeddings of planar graphs

Wikipedia says (in the article on Fáry's theorem),
"Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...

**17**

votes

**3**answers

1k views

### What is Chern-Simons theory expected to assign to a point?

Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a ...

**3**

votes

**2**answers

193 views

### Conjecture of a subset of Wang tile which might be decidable

From the two papers proving the undecidability of Wang tile in 1966 by Berger and in 1971 by RM Robinson, the tiles used in proving undecidability has a general common feature:
The left color and ...

**31**

votes

**0**answers

428 views

### Which region in the plane with a given area has the most domino tilings?

I just finished teaching a class in combinatorics in which I included a fairly easy upper bound on the number of domino tilings of a region in the plane as a function of its area. So this led to ...

**6**

votes

**1**answer

582 views

### What keeps asymptotic Goldbach's conjecture out of reach of current technology?

Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...

**5**

votes

**6**answers

727 views

### practical algorithms for np complete problems

Inspired by:
Conjecture on NP-completeness of tesselation of Wang Tile up to finite size
And the practicality of this topic (solving tessellation on a lattice):
coloring in lattice
Computational ...

**16**

votes

**1**answer

628 views

### Paul Erdős: Determine or estimate the number of maximal triangle-free graphs on n vertices

Among the collections of the open problems of Paul Erdős on the website of
Professor Fan Chung, there is one called "number of triangle-free graphs".
...

**7**

votes

**3**answers

860 views

### Objections to and arguments for the simplicity of all Riemann zeros

It seems to be that the simplicity of all the zeros is quite widely accepted as a working hypotheses, and it is known that a positive proportion are as such.
Titchmarsh explains in the last chapter ...

**4**

votes

**2**answers

302 views

### An integral related to the Euler Gamma function

The question is from the paper http://arxiv.org/abs/1312.7115 (A curious formula related to the Euler Gamma function, by Bakir Farhi): is it possible to express the integral
$$\eta=2\int\limits_0^1 ...

**3**

votes

**1**answer

171 views

### Real analytic ergodic diffeomorphisms of the two sphere

Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$?
(Possibly by perturbing a rotation in the real-analytic topology?)

**7**

votes

**1**answer

509 views

### Ramanujan's problem 754 still open?

In addition to the MO question The Ramanujan Problems. , I would like to ask the following.
Problem 754 from the list of the Ramanujan's problems ( ...

**9**

votes

**0**answers

189 views

### Convergence in $L^2$ of iterated expectations

Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X \in L^2(\Omega,\mathcal{F},\mathbf{P})$.
Define the iterated expectations of X as follows: $X_0 = X$, and, ...

**13**

votes

**1**answer

330 views

### An infinite, amenable, finitely presentable group with no non-trivial finite quotients

My question is a simple one: is there a group with the properties in the title?
In the absence of the 'finitely presentable' hypothesis, an example is provided by Juschenko--Monod's construction of a ...

**25**

votes

**2**answers

910 views

### A group-theoretic perspective on Frankl's union closed problem

Here is a group theoretic phrasing of a special case of the union closed conjecture:
Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half ...

**14**

votes

**0**answers

451 views

### An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...

**17**

votes

**2**answers

737 views

### Floors of rationals to powers: Infinite number of primes?

Let $r=a/b$ be a rational number in lowest terms, larger than $1$,
and not an integer (so $b > 1$).
Q. Does the sequence
$$ \lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor,
...

**4**

votes

**0**answers

208 views

### Stability of analytic Zariski structures

Noetherian Zariski structures are introduced by Hrushovski and Zilber(1996).
An analytic Zariski structure is a generalization of Noetherian Zariski structures, introduced by Zilber and Peatfield.
...

**6**

votes

**0**answers

328 views

### Second duals of Grothendieck spaces

The classical example of a Grothendieck space is $\ell_\infty$. It is also known that its even duals $\ell_\infty^{**}$, $\ell_\infty^{(4)}$, $\dots$ are Grothendieck spaces.
(See, e.g., this note ...

**6**

votes

**4**answers

1k views

### Interactions of number theoretic conjectures and other fields of mathematics

There are many interesting open conjectures in number theory. My question is not about partial results or possible ways to prove them. It is about their interactions with the other fields of ...

**1**

vote

**1**answer

265 views

### Any results towards the irrationality of the sum of reciprocals of perfect numbers? [closed]

This question is a follow up to my comment to Sum of the reciprocal of perfect numbers. I would like to know which results have been published about the possible irrationality of the sum of ...

**16**

votes

**1**answer

688 views

### A Question on 1, 2 ,3 Conjecture

1, 2, 3 conjecture is well-known:
If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the numbers ...

**7**

votes

**4**answers

459 views

### A Special Pair of Models for ZFC (New Version)

Are there two models $M$ and $N$ for $\text{ZFC}$ such that:
(1) $M\subseteq N$
(2) $\aleph_{1}^{N}=\aleph_{1}^{M}$
(3) $\aleph_{2}^{N}=\aleph_{\omega +1}^{M}$
Update: According to Peter's useful ...

**6**

votes

**1**answer

566 views

### A Hot Betting On HOD

Remark: This question is based on an open question at the end of a paper by Hamkins, Kirmayer, and Perlmutter: "Generalizations of the Kunen Inconistency".
$HOD$ as an inner model of $ZFC$ lies ...

**3**

votes

**0**answers

109 views

### A challenging non homogenous fractional inequality

I have posted this question on Stackexchange but it has received no answer so far. It is a challenging generalization of several difficult inequalities, where none of the usual methods used in ...