21
votes
Brute force open problems in graph theory
The existence of a 57-regular "Moore graph" is one such problem.
We define the diameter of a graph $G$ to be the least $l$ such that any two vertices $u,v$ have a path between them using $\...
16
votes
Is there any progress toward solving Gilbreath's conjecture?
There is some theoretical progress towards the conjecture in
Chase, Zachary, A random analogue of Gilbreath’s conjecture, ZBL07808058.
If one models the prime gaps $p_{n+1}-p_n$ (beyond the first gap ...
- 109k
16
votes
Brute force open problems in graph theory
Elaborating on the comment of Wojowu, for what positive integers $q$
does there exist a bipartite graph $G$ with vertex bipartition $(A,B)$
satisfying: (a) $|A|=|B|=q^2+q+1$, (b) $G$ is regular of ...
12
votes
Brute force open problems in graph theory
Question: Does $K_{50}$ decompose into seven copies of the Hoffman-Singleton graph?
The following is copied from https://faculty.math.illinois.edu/~west/openp/hoffsing.html
Definitions: The Hoffman-...
11
votes
How to tackle the smooth Poincaré conjecture
Fastforward $11+\frac12$ years, I thought I'd mention that I tried to do what my original post suggested, which is when I started my PhD and then when I completed my thesis it naturally spawned this ...
- 17.2k
9
votes
5n+1 sequence starting at 7
As far as I understand it, under the current state of things we cannot prove that any orbit of 5k+1 diverges. In fact, as far as I'm aware we cannot even prove that there exists an odd $a>1$ such ...
- 6,100
8
votes
Accepted
Does there exist a comprehensive compilation of Erdos's open problems?
Recently, Thomas Bloom created a website dedicated exactly to this:
https://www.erdosproblems.com/
It currently lists 214 problems, both open and closed. They are all tagged and some problems carry ...
- 13.4k
6
votes
Open problems which might benefit from computational experiments
DeepMind recently announced that their FunSearch methodology was successful at advancing the state of the art of the cap set problem and the online bin packing problem. Under the hood, the code used ...
6
votes
Open problems which might benefit from computational experiments
I have mentioned before the possibility of automated search for bijective proofs. At the time I asked the question, I was imagining a general tool that researchers could apply to any problem of ...
6
votes
Open problems which might benefit from computational experiments
Maybe this can be considered a minor open problem, but you can try to take a look at this Question.
Furthermore, in Section 4 of this preprint of mine Preprint with the general conjecture, I added my ...
5
votes
Integer-distance sets
There has been recent progress concerning finitary integer distance sets in the plane. Greenfeld, Iliopoulou, and Peluse prove that if $P\subset [-N,N]^2$ is an integer distance set not contained ...
- 3,413
5
votes
Accepted
On the number of complete Boolean algebras
The answer is that there are still $2^\kappa$ many isomorphism types of complete Boolean algebras of power $\kappa$.
This is proved by Shelah, see Building complicated index models and Boolean ...
5
votes
Open problems which might benefit from computational experiments
This answer is similar in style to Marco Ripa's concerning a specific preprint. In this preprint by me and Tim McCormack, Weighted Versions of the Arithmetic-Mean-Geometric Mean Inequality and Zaremba'...
4
votes
No starter "accessible" well known open problems
Since I think the question has a reasonable interpretation, let me get the ball rolling with an open problem where I am not aware of any partial progress or proposals for a proof/counterexample.
OPEN ...
4
votes
Accepted
A more complete set of open problems
While looking for sets of open problems, I came across this site for The Association for Mathematical Research. I don’t know much about the organization, but apparently they have a page with sets of ...
- 161
4
votes
Accepted
Goldbach conjecture reformulation
$K$ exists with the required property if and only if the Goldbach conjecture is false.
Assume first that $K$ has the property in the original post. Then $K\geq 6$, and for every prime $q\in[K/2,K]$, ...
- 99.2k
4
votes
Accepted
On the number of values with exactly $k$ prime factors of a given polynomial
It is well-known from sieve theory, in fact such a result can be obtained even if one just uses Brun's original combinatorial sieve, that for any irreducible $f \in \mathbb{Z}[x]$ of degree $d$ that ...
- 25.5k
4
votes
Important open problems that have already been reduced to a finite but infeasible amount of computation
Is there always a prime in the interval $(x^3,(x+1)^3]$ for every natural number $x\geq 2$?
Equivalently the interval may be changed to $[x^3,(x+1)^3]$. Assuming the Riemann hypothesis, this is ...
4
votes
Accepted
3-piece dissection of square to equilateral triangle?
In the paper, "Dissection with the Fewest Pieces is Hard, Even to Approximate" (arXiv, doi) by Bosboom et al., they write:
We have known for centuries how to dissect any polygon $P$ into ...
- 78.6k
4
votes
Books/blogs/websites that have open problems in Algebraic geometry
This book has a "motivating statement" that resonates with the OP:
Open Problems in Arithmetic Algebraic Geometry (2019)
This book originated in the idea that open problems act as
...
3
votes
PhD dissertations that solve an established open problem
Robin Moser in his PhD thesis found a constructive proof for the Lovasz Local Lemma, a problem that was essentially open for decades. This earned him a Godel Prize.
3
votes
Open problems with monetary rewards
The website multimagie.com run by Christian Boyer offers prizes on multiple seemingly elementary problems on magic squares. See the page enigmas for the full list.
I believe the oldest (with a €100 ...
2
votes
Open problems with monetary rewards
$$50000 - each year around the New Year - there is a challenge on combinatorial optimization problems on Kaggle.
For example ongoing (December 2023 - January 2024: "Santa 2023 - The Polytope ...
2
votes
Convex hull in CAT(0)
There is a counterexample if instead of the CAT(0) condition a weaker notion of non-positive curvature is considered. A bicombing on a metric space distinguishes for each pair of points a geodesic ...
2
votes
Accepted
Open problem: $\log n$ factor in Binomial empirical process
See the preprint by
Moïse Blanchard and Václav Voráček, titled "Tight Bounds for Local Glivenko-Cantelli", available here:
https://arxiv.org/abs/2308.01896
.
It clearly explains their ...
- 6,061
2
votes
Brute force open problems in graph theory
There was a question on the math stack exchange a few days ago which essentially asks:
Given a number of edges and vertices, which graph has the maximal number of Hamiltonian paths and how many are ...
1
vote
Could I possibly exploit distinct odd primes raised to 6 to solve Exact Three Cover, when reducing it in Subset Sum?
Going by your notation, $c \in C$, $c=(x_1, x_2, x_3), x_i \in S \simeq \{1, \cdots , N \}$, and $\phi(c) \to \mathbb N$ the map you mentioned, $(\alpha_1,\cdots, \alpha_3) \mapsto \sum_i p_{\alpha_i}...
- 119
1
vote
5n+1 sequence starting at 7
I can't prove that an orbit diverges, but we would expect there to be many because the average growth factor is $\frac{1}{2}(\frac{5}{2})^{2-1} = 1.25>1$. This in contrast to the normal function ...
- 111
1
vote
What is the definition of the function T used in Atiyah's attempted proof of the Riemann Hypothesis?
In his article "The Fine Structure Constant", Atiyah says the following about T on page 6:
For the case of inverse-integer weights, Hirzebruch formalized the notion of exponential
maps and ...
- 537
1
vote
Does pi contain 1000 consecutive zeroes (in base 10)?
Assuming that $\pi$ is a normal number, such a string should exist infinitely many times. However, under the normality assumption, decimal digits asymptotically follow an independent discrete uniform ...
- 388
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