I have a question about the situation of homotopical combinatorics. There are many topics about combinatorial homotopy. But, I can't find any topic about homotopical combinatorics. More precisely, are there any definitions for some combinatorial objects as like as Latin Squares, Designs and etc in homotopy theory?

Do we have any examples about the applications of homotopy theory in combinatorics and graph theory? I think there are some generalizations for the definitions of some combinatorial objects in the language of homotopy theory.

Is this true thinking or not? Please guide me, if you have some experiences.