Homotopical Combinatorics 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.   
 A: There are two papers of M. Wrochna in which ideas from homotopy are used to prove combinatorial theorems:
In http://arxiv.org/abs/1408.2812 he uses ideas from homotopy to give a polynomial-time algorithm for "reconfiguring graph homomorphisms" when the target graph does not contain a $4$-cycle.
In http://arxiv.org/abs/1601.04551 he uses ideas from homotopy to prove a new result on graph homomorphisms which is related to Hedetniemi's Conjecture. Specifically, he proves that all graphs which do not contain a $4$-cycle are "multiplicative." Before this, only one very specific class of graphs was known to be multiplicative (and he also uses homotopy to give a new proof of this result).
A: You might look at the papers of Rade T. Zivaljevic, particularly 
Živaljević, Rade T. Combinatorial groupoids, cubical complexes, and the Lovász conjecture. Discrete Comput. Geom. 41 (2009), no. 1, 135–161. 
and the references there, including a number of homotopical ones. There is an arXiv version of this paper. 
A: Dmitry Kozlov has a book called  Combinatorial Algebraic Topology where he does quite a bit of combinatorial homotopy. I suggest    you have a look   at this book. Maybe it will point you  in the right direction.
