The name of Mathematic Stack Exchange suggests its purpose to provide help answering questions in which OPs are interested, but which are hard to them to answer. But the questions have different importance for OPs. Its motivation can range from an accidental curiosity to a decades long quest for an answer and its value from a solution of a homework task to a research breakthrough.
I propose this tag to support a real research or an application which stuck at a problem. I guess here are many such questions. They may come from non-mathematicians or, at least non-specialists in the question, so it may be too easy to be asked at MathOverflow, but perfectly fit to Mathematic Stack Exchange. I guess that often an asker needs a usual professional consultation. For instance, when a question formulation is simple but out of OP’s grasp, it may be easy for a specialist. But in advanced cases the posed problem may be unknown, so its solution may generate a new topic and the obtained results may be published, which is beneficial both to OP and to an answering specialist. Personally I’m more interested to answer questions relevant to research results or applications (which I answer more readily and devote them more efforts and time) than to explain basics, to verify proofs or to solve con(test) or homework problems (by the way, it is a pity for me that ‘homework’ tag was removed, because I did not like my own homeworks when I was a school boy :-) and I am not going to solve them now, so this tag allowed me automatically ignore such questions).
Thus the proposed tag would allow to an interested specialist to be aware of relevant questions and to track them. Concluding, I’ll cite Karl Popper.
"Yet we also stress that truth is not the only aim of science. We want more than mere truth: what we look for is interesting truth -- truth which is hard to come by. And in the natural sciences (as distinct from mathematics) what we look for is truth which has a high degree of explanatory power, which implies that it is logically improbable. For it is clear, first of all, that we do not merely want truth -- we want more truth, and new truth. We are not content with 'twice two equals four', even though it is true: we do not resort to reciting the multiplication table if we are faced with a difficult problem in topology or in physics. Mere truth is not enough; what we look for are answers to our problems. The point has been well put by the German humorist and poet Busch, of Max-and-Moritz fame, in a little nursery rhyme -- I mean a rhyme for the epistemological nursery:
Twice two equals four: 'tis true,
But too empty, and too trite.
What I look for is a clue
To some matters not so light.
Only if it is an answer to a problem -- a difficult, a fertile problem, a problem of some depth -- does a truth, or a conjecture about the truth, become relevant to science. This is so in pure mathematics, and it is so in the natural sciences”.
As illustrations, I’ll provide a few instances of questions relevant to the proposed tag from my recent Mathematic Stack Exchange experience, which I have already proposed as research topics to my colleagues. The first problem, concerning diagrams constituted by equal rectangles, I commented as follows “It's origin is a computer science problem from a very fundamental level (see the forwarded letter below). I have obtained some partial results and think the question may be a good subject for our joint research, which, I expect, can be published later, because the topic looks new. Although it is specific, it looks like a propitious topic to apply different ideas of graph theory and combinatorics, as I did in my answers (for instance, Euler's formula, Turán's theorem, Erdős–Szekeres theorem, and Kővári–Sós–Turán theorem)”.
A fragment of OP’s letter to me
”I’m currently studying computer science and I came across this question during my research for my bachelor thesis. I reduced the problem that was given to me to a very fundamental level and formulated it in terms of Venn diagram. As this problem seems very hard to solve and I’m unfortunally not a mathematician ... I decided to post it at math.stackexchange. I would like to thank you for your great answers and your efforts - and also for asking your professor as well as your seminar group. As I’m still very interested in this problem I would be very happy if you could inform me about further research so that we can stay in contact. Of course I will also continue to think about it in the future”.
The second problem concerns specific properties of graph colorings and was proposed by a theoretical physicist working in condensed matter and quantum computing, who wrote “I am a tourist of Graph theory but I have this problem I am trying to solve, part of a research objective connected to quantum computing”. Some time ago I have I offered a bounty for this question with the comment
“I thought about this question a bit and I hope it can be answered by forces of MSE users, because it may be not so hard, but it needs different ideas and approaches. Also it gives a unique occasion to find an application of such abstract and combinatorial branch of mathematics as graph coloring is such deep natural science as quantum computing, so I am happy to support it by finding at last a good use for a few of my reputation points. :-) “.