So i was given this question. There are 11 teams in a league. Each team can play against the other team only once. Show that there are always two teams who played exactly the same number of games.
My attempted solution. (based it off an example that was different but also slightly similar)
In a league with two divisions of 11 teams each there is no schedule with each team playing a game with each team in each division. In the language of graph theory, the scheduled games can be viewed as edges with 11 vertices. We are asking for a graph that is regular of degree 11, but we are also asking for the subgraph induced by the 11 teams in one division to be regular of degree 1. Since 1 and 11 are both odd, this is impossible, because every graph has an even number of vertices of odd degree.
We are working on graph theory but I have an assumption that this can be done by pigeon hole principle. I'm slightly confused on how to go about the graph theory method