New Quantitative Bounds for the (p,q)-Theorem for Unions of Convex Sets · arXivDesk2608.13176Aug 13, 202621 pages
New Quantitative Bounds for the (p,q)-Theorem for Unions of Convex Sets
Chaya Keller, Shakhar Smorodinsky
Abstract
A set in Rd is s-convex if it is the union of at most s convex sets. A family F satisfies the (p,q)
property if among any
sets in
, some
intersect. Let
HDd(s)(p,q) be the minimum number of points needed to pierce a finite family of
-convex sets that satisfies the
-property. Alon and Kalai (1995) proved that
HDd(s)(p,q) exists for any
p≥q≥d+1 and any
, but the quantitative bounds they obtained are very loose. We present several improved upper and lower bounds, for a general
and for
-intervals of the line (i.e.,
HD1(s)(p,q) ). In particular, we prove the following: (i) For every
,
and
, if
and
q≥Cdlog(esp) , then
HDd(s)(p,q)≤p−q+1+Od,δ((s⋅qp⋅logqesp)ρd+δ), where
is the exponent in the weak epsilon-net theorem of Rubin (2022). (ii) For
,
p≥q≥2 and
q≥C0slog(2s)log(ep) ,
p−q+s≤HD1(s)(p,q)≤p−q+2s+1 . This result provides the first near-tight estimate for
HD1(s)(p,q) for
. (iii) For any fixed
, there are an integer
κs∈{s,…,2s} and constants
Cs,ps>0 such that, whenever
and
q≥Cslog(ep) ,
HD1(s)(p,q)∈{p−q+κs,p−q+κs+1}. Interestingly, this two-value concentration result holds, although the exact value of the threshold remains unknown. (iv) For any
,
HD3(s)(p,4)≥sp2−o(1) . Already for families of convex sets, this significantly improves the best known lower bound on
HDd(1)(p,d+1) , for all
.