OFFSET
0,3
COMMENTS
a(n) is the minimum number of elements in a poset P such that every poset of size n is isomorphic to a subset of P, where the subset inherits the order from P.
Elementary bounds are a(n) >= 2n-1 because it must contain a chain and an antichain, and a(n) <= 2^n-1 because every partial order embeds into the powerset partial order on n elements. It is shown in the MathOverflow link that a(n) has no polynomial upper bound. This is in particular derived from binomial(a(n),n) >= A000112(n).
a(4) = 8 verified using a computer-assisted proof with a SAT solver.
a(5) = 11 proven on MathOverflow.
a(6) = 16 and 16 <= a(7) <= 25 proven on MathOverflow. - Jukka Kohonen, Jan 15 2021
LINKS
Joel David Hamkins and Fedor Petrov, What is the minimal size of a partial order that is universal for all partial orders of size n?, MathOverflow.
Jukka Kohonen, What is the minimum size of a partial order containing all partial orders of size 5? (proofs of a(5)=11, a(6)=16 and 16 <= a(7) <= 25), MathOverflow.
Caleb Stanford, Alloy program to verify a(n) for small n, GitHub.
EXAMPLE
a(2) = 3 because there are 2 nonisomorphic posets on two elements, and both embed into the poset of three elements {a, b, c} with ordering a < b (and other pairs are incomparable).
a(3) = 5 because all posets on three elements can be embedded into {a, b, c, d, e} with ordering a < d, b < c < d, and b < e.
PROG
(Sage)
# Find an u-poset that contains all n-posets as induced posets.
def find_universal_poset(n, u):
PP = list(Posets(n))
for U in Posets(u):
ok = True
for P in PP:
if not U.has_isomorphic_subposet(P):
ok = False
break
if ok:
return U
return None
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Caleb Stanford, Jan 04 2021
EXTENSIONS
a(6) from Jukka Kohonen, Jan 15 2021
STATUS
approved