How to find volunteer reviewers? I am currently in a little dilemma about publishing a result related to general matching.
The dilemma is, that I am not associated to any research institute and thus do not have contact to professional mathematicians, which I could ask for checking a fairly simple proof for correctness.  

So my question is, how and where I could find an expert, who is willing to take a look at my result and proof. 
  Another question is, what convinces professional mathematicians, that a claimed finding is promising enough to volunteer for reviewing despite limited time and some chance to find a fly in the ointment?  

(Alternatives might be to either ask the MO community for counter examples or, to boldly publish on ArxiV and awaiting the reaction of readers; but it seems to me that neither of these two alternatives is a nice way of getting a review.)  
 A: Answer 1: In universities, research institutions, and the Internet.  Answer 2: nothing does for those professionals who are not interested in the area or specific results, while a well-written abstract and introduction often works for those few who are.
I think you should post your paper in two or more places, so that those who have their own interests in the subject can find it quickly, look it over, and give you their comments.  Nowadays there are many nonprofessionals who are able to provide useful critiques: even if they can't find bugs in your proof, they can find bugs in your exposition, and put pressure on you to improve your presentation.
Regarding MathOverflow, you should be able to review your paper, find the two or three points that are most questionable, and for each point, generate a question for MathOverflow.  Ideally the points will require little technical background and each can form a self-contained question. If necessary, you can link the questions together to form a mathematical narrative.  You can briefly state your motivations and provide a link to the paper.  This way you can get several professionals and amateurs to spend the time where it is needed most (hopefully).
Writing a good question for MathOverflow addressing one of these points is hard.  The benefit from doing such is showing the essence of (that part of) the result.  It can make the rest of the paper easier to write so that it can be easier to read.
Gerhard "Fast, Cheap, Good: Pick Two" Paseman,2016.06.14.
A: This is not an answer to your question, but rather a short relevant observation that is too long for a comment.  I occasionally see math papers with non-standard formatting and am immediately discouraged from reading them.  For example, I might see a paper typeset in Word and will more likely dismiss it than if the identical words were typeset in Latex.  In some respect it's crazy that something as silly as formatting would make a difference as to whether I choose to read a paper.  At the same time, however, use of non-standard formatting can communicate to a reader that the author is an "outsider", unfamiliar with professional mathematics and its norms, raising red flags immediately upon sight.  The reader might never get beyond the title, and the beautiful Word-typeset mathematics might never receive serious consideration.  Although no one in the beginning of the 20th century was using either Word or Latex, I'm sure that the unconventional presentation of Ramanujan's mathematical work likely discouraged many from giving it the proper attention it deserved. 
