Fix ε>0 and a nonnull graph H. A well-known theorem of R\"odl
from the 80s says that every graph G with no induced copy of H contains a
linear-sized ε-restricted set S⊆V(G), which means S
induces a subgraph with maximum degree at most ε∣S∣ in G
or its complement. There are two extensions of this result:
∙ quantitatively, Nikiforov (and later Fox and Sudakov) relaxed the
condition "no induced copy of H" into "at most κ∣G∣∣H∣ induced copies of H for some κ>0 depending on H and
ε"; and
∙ qualitatively, Chudnovsky, Scott, Seymour, and Spirkl recently
showed that there exists N>0 depending on H and ε such that G
is (N,ε)-restricted, which means V(G) has a partition into at
most N subsets that are ε-restricted.
A natural common generalization of these two asserts that every graph G
with at most κ∣G∣∣H∣ induced copies of H is
(N,ε)-restricted for some κ,N>0 depending on H and
ε. This is unfortunately false, but we prove that for every
ε>0, κ and N still exist so that for every d≥0, every
graph with at most κd∣H∣ induced copies of H has an
(N,ε)-restricted induced subgraph on at least ∣G∣−d
vertices. This unifies the two aforementioned theorems, and is optimal up to
κ and N for every value of d.Comment: 11 pages, revised according to the referees' comment