For a positive integer
d, a homogeneous
d-interval is a union of at most
d closed intervals on a fixed line
ℓ. Let
be a system of homogeneous
d-intervals such that no
k + 1 of its members are pairwise disjoint. It has been known that its transversal number
can then be bounded in terms of
k and
d. Tardos [9] proved that for
d = 2, one has
. In particular, the bound is linear in
k. We show that the latter holds for any
d, and prove the tight bound
for
d = 2.