Proof of Large-cube Theorems for Multiverse PL

We need a preliminary definition:

Definition (order-type equivalence) Let N be the nonnegative integers and let k>0 be a fixed integer. Two vectors x=(x1, x2, …,xk) and y=(y1, y2, …,yk) in Nk are order-type equivalent if the set {(i,j)| 1≤i,j≤k, xi < xj} equals {(i,j)| 1≤i,j≤k, yi < yj} and {(i,j)| 1≤i,j≤k, xi = xj} equals { (i,j)| 1≤i,j≤k, yi= yj}.

We also need a version of Ramsey’s Theorem related to order-type equivalence (see Graham, R. L., Rothschild, B. L., and Spencer, J. H. Ramsey Theory, 2nd edition, John Wiley, New York, 1990, page 23).

Ramsey’s Theorem (version) If g is a function on Nk such that Image(g) = {g(z) | z an element of Nk} is finite then there is an infinite set H⊆N such that g is constant on the order-type equivalence classes of Hk. Lemma (Multiverse PL) Let G = (N2, Θ) be any directed graph on the lattice N2.. There is an infinite subset H={h0,h1,h2,…} of N such that for L=H×H the labeling function fL of GL has at most one significant label on L. In fact, the set {fL(z)|z∈L, fL(z)<min(z)} ⊆ {h0}. Proof: We define a function g on N2×N2(which we identify with N4). For each (x,y) in N2×N2, define g(x,y)=0 if (x,y) is not an edge of G and g(x,y)=1 otherwise. By Ramsey’s theorem, there is an infinite subset H of N where g is constant on order-type equivalence classes of H4 (which we identify with H2×H2). Consider the infinite subset L=H×H of N2. We are going to show that GL has at most one significant label on L. Let z be a vertex of GL with significant label fL(z). Let z=x1, x2, …,xt be a path in GL from z to a vertex xt with min(xt)= fL(z). Assume that this path is minimal in the sense that min(xi) > fL(z) for 1≤i<t. Note that g(xt-1,xt)=1 because (xt-1,xt) is an edge of GL (and hence an edge of G). Replace any coordinate of xt that equals min(xt) by h0=min(H) (the smallest integer in the set H) and denote the resulting vertex of GL by w.. Note that (xt-1,xt) and (xt-1,w) are order-type equivalent in N4 and thus g(xt-1,w)=1. Thus, (xt-

1,w) is an edge of GL. Hence, z=x1, x2, …,w is a path in GL from z to a vertex w with min(w)= h0 . By the definition of fL(z), we must have fL(z)= h0 and thus h0 is the only possible significant label for a vertex in GL. We have shown that GL has at most one significant label. In fact, { fL(z)|z∈L, fL(z)<min(z)} ⊆ {h0}.■

Theorem (Multiverse PL) Let G = (N2, Θ) be any directed graph on the lattice N2. Let p>0 be any integer. There are finite subsets E and F such that E⊆F⊂N, |E|=p, and for D=F×F and S=E×E, fD has at most one significant label on S. In fact, { fD(z)|z∈S, fD(z)<min(z)} ⊆ {min(E)}. Proof: There is an infinite subset H= {h0,h1,h2,…} of N such that for L=H×H, {fL(z)|z∈L, fL(z)<min(z)} ⊆ {h0}. This assertion follows from the previous lemma. Let E={h0,h1,h2,…, hp-1}. Define S=E×E. The infinite graph GL has at most one significant label on S, but we want to find a finite set F, E⊆F⊂H, such that GD , where D=F×F, has at most one significant label on S. For each z in S, let z=x1(z), x2(z),…,xt(z) be a path in GL from z to a vertex xt(z) with min(xt(z))= fL(z). For vertices z with fL(z)=min(z), this path consists of a single vertex, z itself. If fL(z) is significant, so h0= fL(z)<min(z),this path may have arbitrary but finite length and involve vertices in L but not in S. Since S is finite, we can choose a set F, E⊆F⊂H, such that D=F×F contains the vertices of all such paths. For this choice of F, fD has at most one significant label on S. In fact, {fD(z)|z∈S, fD(z)<min(z)} ⊆ {h0}={min(E)}.■ NOTE: In the examples TL and SL, we shall assume the graph G satisfies the condition that if (x,y) is an edge of G then max(y) < max(x). With this “downward” condition on edges of G, we can take D=S in the above theorem. In fact, we could take D=S if we assumed max(y) ≤ max(x). HIGHER DIMENSIONS: The above results and proofs extend in a straightforward way to graphs G on Nk.