# MO-Hard Questions

The term "MO-Hard" has entered the mathematical vocabulary, describing interesting questions that were asked on MathOverflow but haven't been answered in spite of efforts by many community members. Each one of these is a MathOverflow success story! Open problems are precious commodities that have driven mathematical research for centuries. While these are not necessarily at the same level as the Riemann Hypothesis and the Jacobian Conjecture, we should celebrate them and promote them!

While the unanswered tab is a good way to find such gems, questions that have interesting partial answers do not appear there. In the same spirit as Best of MathOverflow, let's collect MO-Hard questions here. Questions that still appear in the unanswered list are welcome since they might disappear any day. Comments, anecdotes and other remarks are also encouraged.

• Francois, could you clarify how this is different from the MO question looking for "open problems that are easy to state but seem beyond current technology"? – Yemon Choi Aug 5 '14 at 21:06
• I would also debate the premise that "Open problems are precious commodities that have driven mathematical research for centuries", unless it is prefaced by the word SOME – Yemon Choi Aug 5 '14 at 21:07
• In the next 24 hours, I will make a meta post regarding promoting MathOverflow at ICM. This question addresses part of one of the aspects; for my purpose it would help to associate with such questions a number corresponding to one of the areas icm2014.org/en/program/scientific/topics covered by the Congress. While this list should be conference-independent, I would appreciate a large and diverse list to show in Seoul next week: adding the appropriate area tag would help me, even if the tag is not a number. Gerhard "Promoting MathOverflow at ICM2014 Seoul" Paseman, 2014.08.05 – Gerhard Paseman Aug 5 '14 at 21:07
• @YemonChoi: I would love to clarify but I fail to see the similarity since I'm asking for unanswered MO questions. Maybe I should clarify that MO-hard does not mean open? In fact, I commonly heard the term in a sentence like: "this might not be open but it is at least MO-hard..." – François G. Dorais Aug 5 '14 at 21:39
• That clarifies - I misread part of what you wrote, while skimming hastily – Yemon Choi Aug 5 '14 at 21:41
• This question is still unanswered but led to a paper in the PNAS. – Bill Johnson Aug 6 '14 at 18:02
• I don't understand why Benjamin Dickman's mention of his question "Probability that a stick randomly broken in five places can form a tetrahedron" was downvoted and then deleted. The downvote was not explained. – Joseph O'Rourke Aug 7 '14 at 23:09
• @JosephO'Rourke I deleted it after the down-vote (the original MO post also received its second down-vote earlier today...) as I was concerned the mention was perceived as something too close to self-promotion (or otherwise off-topic). I've had several unexplained down-votes on MESE recently, too, on what I thought were innocuous answers. In any event: I've "undeleted" the meta.MO answer for the time being. – Benjamin Dickman Aug 7 '14 at 23:50
• @BenjaminDickman: Each author of their own open problem is best placed to post in this thread. I think such posting does not constitute "self-promotion." Or if it does, I have trumped you $4 {\times}$. :-) – Joseph O'Rourke Aug 7 '14 at 23:52
• This question of mine hasn't been answered: Is there a (discrete) monoid M injecting into its group completion G for which BM is not homotopy equivalent to BG? – Omar Antolín-Camarena Nov 7 '14 at 0:07
• What should be done when one of the questions listed here is resolved? Should the meta answer be deleted? edited to add a note? – Nate Eldredge Jun 24 '16 at 16:46
• @NateEldredge consider adding an answer here: meta.mathoverflow.net/questions/617/best-of-mathoverflow – François G. Dorais Jun 24 '16 at 21:35
• I have voted to close this question since it seems to attract low quality arguments from people who are not interested in real mathematics. – Joseph Van Name Mar 21 '19 at 12:49

Here is a prime example of a MO-Hard question that doesn't appear on the unanswered list.

Polynomial bijection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$?

The title says it all. In fact, the body of the question is just one line. It has had 16 answer attempts though most have been deleted after an error was found...

• One might add to the title question that has transpired in the comments, a “polynomial injection from $\mathbb Q\times\mathbb Q$ to $\mathbb Q$” problem also seems hard. – Emil Jeřábek Aug 5 '14 at 19:42
• "most have been deleted after an error was found" -> Where should "nice try" attempts for open problems settle? Maybe they shall be still around somewhere to give hints for ones who want to try the problem? – Vi. Oct 11 '14 at 1:19
• Currently 17 answers, of which 16 deleted. – Gerry Myerson Dec 7 '16 at 4:24

This question is a very nice topological one, a cousin of Brouwer's fixed point theorem and related to several questions in the literature:

Suppose that $f$ and $g$ are two commuting continuous mappings from the closed unit disk to itself. Does there always exist a point $x$ such that $f(x)=g(x)$?

The question Polynomial representing all nonnegative integers asked by Poonen has been on MO for a very long time. (It should possibly be mentioend that it was asked elsewhere earlier, in particular on Poonen's website.) It received various interesting answers, but none of them solved the problem.

The question is short so I reproduce it:

Is there a 2-variable polynomial $f(x,y) \in \mathbf{Q}[x,y]$ such that $f(\mathbf{Z} \times \mathbf{Z})=\mathbf{N}$?

For $3$ (and more variables) it is well known such a polynomial exists (basically going back to Gauss).

• Temporary comment added on Gerhard Paseman's request: I'd classify that under 3 (Number Theory) though it might touch upon other things too. – user9072 Aug 5 '14 at 22:17

I make no claim to the significance of any of these problems, nor that they are even "hard"—some are too peripheral to attract interest. With those caveats, here are a few of mine (in chronological order):

• "known to be inaccessible to the experts", or "known to the experts to be inaccessible"? – Gerry Myerson Aug 9 '14 at 12:13
• @GerryMyerson: Thanks for the correction---rather different senses! – Joseph O'Rourke Aug 9 '14 at 12:25
• "Known to inaccessible experts" – Will Jagy Nov 7 '14 at 21:02
• domotorp spelled backwards is Proto-Mod. – Gerry Myerson Jan 12 '15 at 6:49

I posted a question about a year ago on MO (and even further back on MSE) that remains elusive:

Randomly break a stick in five places.

Question: What is the probability that the resulting six pieces can form a tetrahedron?

The two breaks to form a triangle problem has a collection of proofs on MO here; the five breaks for a tetrahedron problem is much harder, and relates to the Cayley-Menger determinant.

Another one is my question about whether a particular manifold admits a Riemannian metric.

Ultrafilters and automorphisms of the complex field

whether the existence of non-principal ultrafilters (over the natural numbers) suffices to imply the existence of a nontrivial automorphism of the complex field $\mathbb C$. In set theoretic terms, the question is whether (under appropriate large cardinal assumptions) there is such an automorphism in $L(\mathbb R)[\mathcal U]$ where $\mathcal U$ is a nonprincipal ultrafilter on $\mathbb N$.

• This has been answered by Paul Larson. – Douglas Ulrich Nov 11 '18 at 20:17

One I like very much is

Can we color $\mathbb Z^+$ with $n$ colors such that $a, 2a, \dots, na$ all have different colors for all $a$?

Among the failed proposed attempts, the ones that appeared most promising were in essence group- or number-theoretic, for instance the question of whether the partial graph of multiplication $$\{(a,b,c)\in\{1,\dots,n\}^3\mid ab=c\}$$ could be extended to a group operation on $\{1,\dots,n\}$, with the first counterexample being $n=195$.

A question that fascinates me is the problem of characterizing the subsets $S$ of the unit circle $\mathbb T$ for which there is a power series with radius of convergence $1$ that, on $\mathbb T$, converges on $S$ and diverges otherwise.

(The question has also been asked on MSE a couple of times.)

• While an undergraduate, I once asked my lecturer about this, and he said "in my view there is no reasonable classification". (I see from your answer on that MO question that he has not changed his mind) – Yemon Choi Aug 5 '14 at 21:05

I came across this question the other day...

Order-increasing bijection from arbitrary groups to cyclic groups

... It's possible that the question has been answered in the literature by now, as it is a little old. I don't know much about it but the discussion is very interesting, and suggests that a solution would be of some significance.

• This question now appears in the 18th ed. of the "Kourovka notebook" as Problem 18.1 (suggested by I. M. Isaacs). I think it was also on mathoverflow where it was pointed out first (by Isaacs) that a "solution" in the literature is not complete. – Frieder Ladisch Aug 22 '14 at 10:04
• A solution to this question has appeared on the arXiv: arxiv.org/pdf/1812.04167.pdf – Nick Gill May 7 '19 at 11:41

Even though it has not been here for particularly long, I think it is fair to add Is the set $AA+A$ always at least as large as $A+A$? to this list.

It is a problem that at first glance seems very simple, but the partial answers so far are revealing that there might be something deep going on.

Here are my MO unsolved questions :

Intersecting Family of Triangulations

This is an Erdos-Ko-Rado type conjecture for triangulations. It was largely extended by Gjergji Zaimi to polytopes without triangular faces.

A Weak Form of Borsuk's Conjecture

Can we cover a convex polytope $P$ with $m$ facets by $m$ (or $poly(m)$) sets of smaller diameter?

A curious generalization of Helly's theorem

Helly-type theorems for pairs of convex sets.

Infinitely many primes, and Mobius randomness in sparse sets

Is it possible to prove Mobious randomness or PNT for a subset $A$ of integers with $|A \cap \{1,2,\dots,n\}| \le n^{1/2-\epsilon}$?

The next items appeared as open questions earlier so I knew they are not easy but I thought they had a chance.

Volumes of Sets of Constant Width in High Dimensions

Can we have sets of constant width with exponentially (in the dimension) smaller volumes than balls? (Oded Schramm)

A Combinatorial Abstraction for The "Polynomial Hirsch Conjecture"

The shortest path in first passage percolation Update: solved by Daniel Ahlberg and Christopher Hoffman.

Optimal Monotone Families for the Discrete Isoperimetric Inequality

This question is a relative to the well-known question of the chromatic number of unit-distance graphs (so, thematically speaking, it lies between geometry and combinatorics with a Ramsey flavor). There are in fact several questions in there, please see the original question.

This question from Riemannian geometry:

Consider a two-dimensional sphere with a Riemannian metric of total area $4\pi$. Does there exist a subset whose area equals $2\pi$ and whose boundary has length no greater than $2\pi$?

The question that asks if a square of area $4n^2$ is the geometric figure with the most number of domino tilings, of all figures of the same area, here, is still unsolved. I find this question quite intriguing.

Self-promotion, but I would like to mention my question Meager subspaces of a Banach space and weak-* convergence. It contains two questions. I have resolved Q1 and posted it as an answer (hence the question is no longer "unanswered") but Q2 has not been resolved. I quote it here for visibility.

Q2. Let $X$ be a Banach space. Let us say a linear subspace $E \subset X$ determines weak-* convergence (of sequences) if for every sequence $\{f_n\} \subset X^*$ such that $f_n(x) \to 0$ for every $x \in E$, we have $f_n(x) \to 0$ for every $x \in X$. Is every such $E$ nonmeager?

The converse is an easy exercise with the uniform boundedness principle.

Update Q2 has now been resolved. The answer is No (though I would still be interested in a separable counterexample). Should I delete this?

• @FrançoisG.Dorais commented that MO-hard does not imply open. – LSpice Sep 5 '17 at 22:03

There are several MO-hard problems about smooth and proper schemes over $\mathbb Z$, including:

Smooth proper schemes over $\mathbb Z$ with points everywhere locally

Non-simply-connected smooth proper scheme over $\mathbb Z$

What can be the dimension of a pointless smooth proper $\mathbb Z$-scheme?

All these were inspired by Poonen's (solved) question, asking whether such a scheme necessarily has a section:

Smooth proper scheme over $\mathbb Z$

This question describes a combinatorial game based on commutative algebra where the game state is a ring and a move is taking the quotient by a nonzero element:

A Game on Noetherian Rings

The hard problem is to either determine which positions are winning and which are losing, or to show that the previous problem has no solution.

Three of my MO questions appear to be quite hard:

1. The existence of good covers of topological manifolds. In addition to MO, I asked Ciprian Manolescu (specifically, the 4-dimensional case), he did not know.

2. Open immersions of open manifolds. In addition to MO I asked Yasha Elishberg: he does not know but expects a negative answer.

About discontinuties of a function and its Fourier transform : Is this statement which relates the Fourier transform of a function to its singularities correct?

• Some illustrations added : mathoverflow.net/q/165038/14414 – Rajesh Dachiraju Aug 19 '14 at 9:17

I think that this question is may be non easy. it is about a certain generalization of the Poincare Bendixon theorem

A certain generalization of the Poincare Bendixon theorem

Here are some number theory questions that (so far as I know) are original with me and appeared on MathOverflow first . They may be hard; at this writing none of them are satisfactorily answered.

Jumping prime sieve: How do these primes jump?

Playing prime leapfrog: Playing leapfrog with primes

Semiprime cover: How much do these interval collections cover?

Combinatorial ring toss: Has anyone seen this version of ring toss (combinatorial object) before?

Square difference sequences: How many sequences of rational squares are there, all of whose differences are also rational squares?

Gerhard "Will Accept Fame From Questions" Paseman, 2017.09.02.

This question entitled "Limit cycles as closed geodesics in negatively or positively curved space" is not answered yet. So may be the above question and its linked question entitled "Limit cycles as closed geodesics(2)" can be counted as a non trivial question.

All of my 25 questions (except for one) from July 19, 2016 to now which is October 24, 2017 are still unanswered (and more are on the way). These questions have covered diverse topics such as forcing, braid groups, large cardinals, cellular automata, linear algebra, and cryptography but the main these of these questions is the elusive notion of self-distributivity.

• OP is asking for "questions that were asked on MathOverflow but haven't been answered in spite of efforts by many community members" and "questions that have interesting partial answers." OP didn't just write, "list all the unanswered questions on MO" – that would be a pointless exercise. What makes one or more of your 24 questions stand out from all the other unanswered questions? – Gerry Myerson Oct 24 '17 at 11:38
• The point is that the main theme of most of these diverse questions is the notion of self-distributivity. Mathematicians are comfortable with the notion of associativity. Mathematicians everywhere study objects with some form of associativity to the point of monotony. On the other hand, mathematicians are not comfortable with the notion of self-distributivity and inner endomorphisms (except when it comes from groups). Or do we just only want to study the associativity property on this site? – Joseph Van Name Oct 24 '17 at 12:11
• And how is one supposed to know how much effort people have spent on a certain problem? What if a problem is so intractible with there current mathematical knowledge that people simply do not know where to begin attacking the problem? – Joseph Van Name Oct 24 '17 at 12:19
• All of my questions are valuable problems to work on since they are some of the most natural questions that come up from the theory of self-distributivity. Instead of downvoting this answer, why don't you attempt to be a productive citizen and try solving one of them? – Joseph Van Name Mar 21 '19 at 23:43
• The 'mathematics' community is deeply mistaken for considering associative algebraic operations that are merely rehashes of the Kindergarten operations +,-,*,/ and the composition operation $\circ$ if you are a group theorist. – Joseph Van Name Mar 22 '19 at 2:05