Using Topological Methods to Force Maximal Complete Bipartite Subgraphs of Kneser Graphs

Santa likes to run a lean and efficient toy-making operation. He also likes to keep up with the latest math developments. So naturally, when Santa was building a new factory a few years back, he designed an interesting factory and method for assigning workers to toys that was based on a recent article he read in the Monthly ??. Here were some of Santa’s constraints. As it turns out, each toy requires n elves, but some toys take longer than other to make. There is one station for the fabrication of each type of toy. Due to a powerful Elf Workers Union, elves are allowed to take breaks of indeterminate length between projects, but they generally don’t dawdle for too long because they are not paid for their break time. Also, each worker was to be initially trained in making certain toys in conjunction with certain groups of elves. After reading ??, Santa envisioned building we he calls a Kneser factory, which would have the propert that (i) that each n-subset of elves would be trained to make certain toy, and (ii) disjoint n-subsets of elves are never assigned to the same toy-making station. This would ensure that whenever n elves came back from their break and were ready to work, they could be immediately assigned to a toy-making project at an available station. The first question Santa asked was: