Tverberg's theorem for unions of convex sets: Sharp bounds and colored extensions · arXivDesk
2607.12449Jul 14, 202616 pages. Note added: Keller and Smorodinsky independently proved Theorem 1.5 in broader setting via different method. arXiv:2607.10496
Tverberg's theorem for unions of convex sets: Sharp bounds and colored extensions
. A recent breakthrough of Alon and Smorodinsky established the first effective upper bounds
fr(d,s,…,s)≤Cdr2srlogrlog(esr)
for this problem. We obtain an asymptotically sharp lower bound by proving
fr(d,s,…,s)≥c(d−r+2)srlog(s+1)
for every
d≥r+2
, which shows that
fr(d,s,…,s)=Θd,r(srlogs)
for every fixed
d≥r+2
. We also prove the general lower bound
fr(d,s,…,s)>smin{d,r}
. On the other hand, we develop a local counting argument to show that
fr(d,s,…,s)≤Cdrsrlog(ersr)
and
fr(d,s,…,s)≤Cdrd+2sd+1log(ers)
whenever
r≥d+1
, improving the upper bound of Alon and Smorodinsky. We also study two colored analogues. The direct Bárány--Larman-type extension, in which one seeks
r
disjoint rainbow sets chosen from
d+1
color classes, fails as soon as two convex pieces are allowed. Nevertheless, we identify the correct colored formulation and prove a complete transversal theorem with quantitative bounds, which was also independently obtained by Keller and Smorodinsky.