We say (n, e) → (m, f), an (m, f) subgraph is forced, if every n-vertex graph of size e has an m-vertex spanned subgraph with f edges. For example, as Turán proved, (n, e) → (k, (k2)) for e > tk-1 (n) and (n, e) → (k, (k2)), otherwise. We give a number of constructions showing that forced pairs are rare. Using tools of extremal graph theory we also show infinitely many positive cases. Several problems remain open.
- Ramsey's theorem
- Turán's theorem
ASJC Scopus subject areas
- Theoretical Computer Science
- Discrete Mathematics and Combinatorics