We look at such posets whose assigned real number is largest as compared to other such posets with the same field. We obtain results about the structure of such optimized posets using large cardinals in an essential way. There are a number of new technical ideas needed to prove these results, and I am at the outer limit of what I can safely claim without a detailed manuscript. These results now look stable enough for publication. We begin by fixing standard terminology and notation regarding posets.
Finite range means that the range is a finite set.
And here is a sharper form. First we begin with two semifinite forms. The hypotheses are infinite but the conclusion is finite. Here we will not need the parameter r. And here is a sharper form.