A340318 Minimum size of a partial order that contains all partial orders of size n.

0, 1, 3, 5, 8, 11, 16

Offset: 0

Author

Caleb Stanford, Jan 04 2021

Keywords

• nonn
• more

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

Examples

			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.
		

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.

Crossrefs

Cf. A000112, A001035, A004401, A097911.

Programs

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

Extensions

a(6) from Jukka Kohonen, Jan 15 2021