Hitting all maximum independent sets

We describe an infinite family of graphsGn, where Gn has n vertices, independence number at least n/4, and no set of less than √ n/2 vertices intersects all its maximum independent sets. This is motivated by a question of Bollobás, Erdős and Tuza, and disproves a recent conjecture of Friedgut, Kalai and Kindler. Motivated by a related question of the last authors, we show that for every graph G on n vertices with independence number (1/4 + ε)n, the average independence number of an induced subgraph of G on a uniform random subset of the vertices is at most (1/4+ε−Ω(ε2))n. 1 Background and results The following conjecture appears in a recent paper of Friedgut, Kalai and Kindler. Conjecture 1.1 ([8], Conjecture 3.1). For every α ∈ (0, 1/2) there exists k and τ > 0 such that if G is a graph on n vertices with maximum independent set of size αn, then there exist pairwise disjoint subsets of vertices A1, A2, . . . , Ar in G such that 1. |Ai| = k for all i. 2. | ∪i=1 Ai| ≥ τn 3. Every maximum independent set in G intersects every set Ai. Here we show that this is false for every fixed α ∈ (0, 1/2) even if the requirement (2) is omitted, but that the assertion does hold for any α > 1/2. We also discuss several related problems. For a graph G = (V,E) let h(G) denote the minimum cardinality of a set of vertices that intersects every maximum independent set of G. Bollobás, Erdős and Tuza (see [7], page 224, or [5], page 52) raised the following conjecture. ∗Princeton University, Princeton, NJ 08544, USA and Tel Aviv University, Tel Aviv 69978, Israel. Email: nalon@math.princeton.edu. Research supported in part by NSF grant DMS-1855464, BSF grant 2018267 and the Simons Foundation.