Add field to posts (checkbox) that only one post can have

I am trying to add a field to WordPress posts (not pages) which would simply include a label and a checkbox. Something like: Make this the sticky post? If check, the key should be set to TRUE or something that is easily accessible. The problem is, however, that I only want one post to have possibly the TRUE value. This means, when I have checked the field in post-1, and afterwards I check that field in post-2, post-1’s field value should be set to FALSE so that only one post can have TRUE at the same time.

Creating Ajax backend for Specialised Page Template? Should admin-ajax.php be used?

We’re scoping an upcoming project that will require the development of a large php web application within an existing wordpress site – this is for our use only and we’ve no intention of trying to package it for other users; so interoperability isn’t really a concern as long as we can continue to apply WordPress updates as they’re released without breaking our application.