Could you please clarify your definitions via the example polygon below?
Is $\{a,b,c,1,12\}$ an antirectangle of size $5$?
Is this a maximal antirectangle?
What is a largest antirectangle that includes a corner in this example?
Again, just illustrating the definitions, here is a maximal antirectangle, following the OP's comments:
![AntiRect2][2]