Let be a -uniform -regular hypergraph, so that for each there are exact many having .
Use the probabilistic method to prove: For , there is a 2-coloring such that does not contain any monochromatic hyperedge .
Let be a -uniform family, and suppose that it satisfies that for any .
- Fix any . Show that the family is an anti chain.
- Show that .
Given a graph , a matching is a subset of edges such that there are no two edges in sharing a vertex, and a star is a subset of edges such that every pair of distinct edges in share the same vertex .
Prove that any graph containing more than edges either contains a matching of size or a star of size .
Let and let be a -uniform family such that for all . Show that .