Efficient algorithms for near-optimal independent sets in random graphs

Determine whether there exists an efficient algorithm that, given a random graph on n vertices with average degree d, finds an independent set of size (1+ε)(n/d)\log d for arbitrarily small fixed ε>0.

Background

The paper connects the difficulty of improving Shearer’s bound with an algorithmic problem. Although random graphs are known to contain many independent sets of the target size, existing results rule out local algorithms for finding them, leaving the existence of efficient algorithms in a broader class unresolved.

References

For example, it is a major open problem to determine if there is an efficient algorithm that can find an independent set of size $ (1+)(n/d)\log d$ in the random graph on $n$ vertices of average degree $d$, despite the fact that we know there are many such independent sets.

— A new lower bound for the Ramsey numbers $R(3,k)$  (2505.13371 - Campos et al., 19 May 2025) in Section 1, subsection “The asymptotic value of R(3,k)”