Bollobás–Erdős–Tuza conjecture for graphs with linear independence number

Prove that for every fixed constant c in (0,1], every n-vertex graph G with independence number alpha(G) at least cn has a hitting set for its maximum independent sets of size o(n).

Background

For a graph G, h(G) denotes the minimum size of a vertex set intersecting every maximum independent set. The paper identifies the Bollobás–Erdős–Tuza conjecture as the central open question: positive linear independence ratio should force a sublinear hitting set. The conjecture is known above the threshold alpha(G)>n/2 by Hajnal’s intersection–union inequality, but no general sublinear bound is known below that threshold.

The paper develops reductions showing that resolving the conjecture for dense regular graphs of any fixed positive linear degree ratio, or for highly connected graphs within a hereditary class, is asymptotically equivalent to resolving it in the unrestricted setting. It also records constructions with h(G)=Omega(sqrt(n)), demonstrating that a bounded hitting set cannot hold in full generality.

References

The central open question is the following conjecture, posed at the 1991 Visegrád conference and later recorded by Erdős; see . For every fixed $c\in(0,1]$, every $n$-vertex graph $G$ with $\alpha(G)\ge cn$ satisfies $h(G)=o(n)$.

— Hitting Maximum Independent Sets in Dense and Highly Connected Graphs  (2608.18963 - Bai et al., 19 Aug 2026) in Conjecture 1.1, Section 1 (Introduction)

For each fixed $0<\rho\le1/3$, does every $3$-colorable graph $G$ of order $n$ with $\kappa(G)\ge\rho n$ satisfy $h(G)=o(n)$? Equivalently, can one obtain, for every such fixed $\rho$, a function $f_\rho(n)=o(n)$ such that $h(G)\le f_\rho(n)$ for all eligible $G$?

— Hitting Maximum Independent Sets in Dense and Highly Connected Graphs  (2608.18963 - Bai et al., 19 Aug 2026) in Problem 2, Section 6 (Open Problems)