Large Independent Sets from Local Considerations

Abstract The following natural problem was raised independently by Erdős–Hajnal and Linial–Rabinovich in the early ’90 s. How large must independence number $$\alpha (G)$$ α ( G ) of a graph G be whose every m vertices contain an independent set size r ? In this paper, we discuss new methods to attack problem. first approach, based on bounding Ramsey numbers certain graphs, allows us improve previously best lower bounds due Linial–Rabinovich, Alon–Sudakov. As example, prove that any n -vertex having 3 among 7 has (G) \ge \Omega (n^{5/12})$$ ≥ Ω n 5 / 12 . This confirms conjecture Erdős Hajnal should at least $$n^{1/3+\varepsilon }$$ 1 3 + ε brings exponent halfway possible value 1/2. Our second approach deals with upper bounds. It relies reduction original question extremal What is minimum 2-density (The 2- density H defined as $$m_2(H):=\max _{H' \subseteq H, |H'| 3} \frac{e(H')-1}{|H'|-2}$$ m 2 H : = max ′ ⊆ , | e - ) no previous Krivelevich Kostochka-Jancey. part our arguments, link other well-studied questions. leads many interesting directions for future research.