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Results tagged with open-problems
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user 2233
If it turns out that a problem is equivalent to a known open problem, then the open-problem tag is added. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it before, and with what results?"
6
votes
Accepted
Is the Manickam-Miklós-Singhi Conjecture solved?
I am aware of the paper, but I am not sure that MO is the right forum for this sort of question. Nonetheless, let me try to provide some information in as neutral a manner as possible.
Note that t …
- 32.1k
23
votes
Difficult examples for Frankl's union-closed conjecture
Here is a nice example due to Bjorn Poonen, which I have taken from this survey paper of Bruhn and Schaudt. It is motivated by the following observations. Let $\mathcal{A}$ be a union-closed family. …
- 32.1k
44
votes
5
answers
23k
views
Does pi contain 1000 consecutive zeroes (in base 10)?
The motivation for this question comes from the novel Contact by Carl Sagan. Actually, I haven't read the book myself. However, I heard that one of the characters (possibly one of those aliens at th …
- 32.1k
29
votes
Important open problems that have already been reduced to a finite but infeasible amount of ...
One of the most important open questions in graph theory is Hadwiger's conjecture, which asserts that every graph with no $K_{t}$-minor is $(t-1)$-colourable. The cases $t=1,2$ are trivially trivial, …
4
votes
PhD dissertations that solve an established open problem
Maria Chudnovsky's PhD thesis gives a proof of the Strong Perfect Graph Conjecture. This is a conjecture of Claude Berge from 1961 (hence meets the 25 year criterion), and was considered one of the h …
14
votes
Current state of the Komlos conjecture on vector balancing
As far as I know, the state of the art is that one can actually achieve $K=K(n)=O(\log^{\frac{1}{2}} n)$ independent of the dimension $d$. Moreover, one can actually find the weights $w_i$ via an eff …
- 32.1k