Well, a personal anecdote, I worked with a famous guy for some years. His basic strategy was to mail, on paper, anything publishable in a draft to one or two dozen parties that might be presumed to be interested. If nobody replied inside a month he submitted it somewhere. The point, in my mind, was that if one other person sees your stuff early you may get robbed, but if 20 see it early they are all witnesses. Later they came up with the arXiv.
The other, well-known side, is that if you share your stuff with the top expert in the field, that person may send you back a note saying "that was fun, here is the answer" and promptly forget all about it. You have not been cheated but there is still a problem.
EDIT There seem to be mixed impressions of what I meant in the preceding paragraph, and for whom the situation would remain a problem, so maybe I had better describe my own experience again. I have told this story many times, with names, and I think the story only does people credit, but I think on MO I ought to stick to anonymity. Email me if you want more detail. In graduate school I was working on minimal submanifolds. My adviser came up with a fairly specific problem, suggested I work on it, and asked one or two guys in the same department if they thought it was new, which they did. It still took me some time but I was getting there. My adviser was away somewhere giving a talk, and, once again, mentioned the problem to a guy. The difference was that this guy is a leading light in similar problems, went home, solved my dissertation problem in one evening on some 30 pages of notes, and put those in a drawer and forgot all about it. But at some point he happened to mention to my adviser that it was a good problem, he had completely solved it. My adviser mentioned this to me, and I was terrified. How could I submit this as a dissertation if this other guy solved it already? At some point I contacted him, he said, don't worry, it's your problem, I don't need it, you just finish it up and it's your dissertation. Finally, after I finished, I did ask to see his notes, he found them eventually and sent me copies, but even between him and his adviser at the 1992 Park City summer program no sense could be made of the notes by anyone concerned.
So I suppose I would say, along with Kevin's comment, that the nature and severity of the "problem" when the world champion in your area solves your problem in an afternoon (but has not the slightest intent to publish, ever) depends on your position and how much you need this as a publication/dissertation and how critical it may be that the work be perceived as your own and original. I may have misunderstood my position in graduate school, and everybody behaved well in my opinion, but it was certainly scary. I think I do see Kevin's point that, as a journal referee, he is often confronted by work that is already known, "in the air" as they say, or where the most likely techniques are pretty obvious as soon as the statement of the theorem is read, but he will still accept it for many journals.
I think it is fair to say people pick and choose what of their stuff to put on MO. This is probably healthy. We should struggle rather than getting handed everything.
Given that this question is Country and Western, I am taking this opportunity to point out that I went to high school with Paul Ginsparg, founder of the arXiv. He was a year older. It is a good bet that he is still a year older. Also Natalie Portman.
http://en.wikipedia.org/wiki/Syosset_High_School
http://en.wikipedia.org/wiki/Paul_Ginsparg
http://en.wikipedia.org/wiki/Natalie_Portman
Best Answer
I'd choose $R(5,5)$ and other unknown Ramsey numbers, since