Sizes of witnesses in Covtree

Abstract: Given a set $Γ$ of $k$ unlabelled posets, each of size $n$, we say that a poset $Q$ is a \emph{witness} to $Γ$ if $Γ$ is the set of downsets of size $n$ of $Q$. We say that $Q$ is a \emph{minimal witness} if it does not contain a proper downset that is itself a witness to $Γ$. Motivated by the causal set approach to quantum gravity, we study the upper bound on the size of minimal witnesses as a function of $n$ and $k$. We show that there is no linear upper bound of the form $n+k+c$ for any constant $c$. We introduce the \emph{exchange graph of downsets} as a new tool to study this scenario, and use it to show that all minimal witnesses $Q$ satisfy the bound $|Q|\leq nk-n$, and that when $k=3$ there is at least one minimal witness $Q$ that satisfies the bound $|Q|\leq \frac{3}{2}(n+1)$.

• The paper establishes that no linear upper bound of the form n+k+c exists for minimal witness sizes in covtree.
• It employs exchange graph techniques to derive improved bounds, including a sharp bound for k=3 cases.
• Explicit constructions using Newtonian, tall, and height-2 poset families reveal unbounded gaps relative to naive expectations.

Sizes of Witnesses in Covtree: Structure, Bounds, and Combinatorial Barriers

Introduction and Motivation

This paper addresses a fundamental question in the combinatorial structure underpinning the covariant tree (covtree) framework for poset growth, motivated by the causal set approach to quantum gravity. In covtree, nodes are sets of unlabelled posets representing possible “histories” of spacetime at a given cardinality, and the existence and size of witnesses—posets from which every node’s constituent downsets of fixed size $n$ can be obtained uniquely—encodes critical structural constraints. The focus here is to determine upper bounds on sizes of minimal witnesses $Q$ to a given set $\Gamma_n$ consisting of $k$ distinct unlabelled posets of size $n$, and to characterize when such upper bounds can or cannot be given in terms of simple linear formulae. This is deeply relevant for both theoretical physics, where label-independence and discrete growth models play a role in the causal set programme, and pure combinatorics, where understanding the enumeration and interrelations among posets is a significant foundational issue.

Covtree Structure and Witnesses

Covtree is constructed so that each node at level $n$ comprises all sets $\Gamma_n$ of unlabelled posets potentially obtainable as the collection of downsets of size $n$ from some finite poset $Q$. If such $Q$ exists, it is called a witness for $Q$0; if, in addition, $Q$1 contains no proper downset that is itself a witness to $Q$2, $Q$3 is minimal. Brute-force combinatorial construction of covtree is challenging due to the rapid growth of the number of posets with cardinality. While every $Q$4 that is a node may have many minimal witnesses—possibly of varying sizes—understanding the minimal possible size is critical, both for feasible enumeration and to elucidate structural obstructions.

The basic architecture of labelled versus unlabelled poset possibilities is shown below:

Figure 1: The tree of labelled posets, illustrating the branching of possibilities induced by sequential growth protocols.

The first three levels of covtree, emphasizing the coalescence of labelled possibilities into unlabelled nodes, are depicted here:

Figure 2: The first three levels of covtree, demonstrating the identification and merging of labelled paths up to isomorphism.

Upper Bounds and Obstructions for Witness Size

Known Results and Simple Bounds

From previous results, it is established that if some witness exists for $Q$5 of size $Q$6, then there is always a witness of size at most $Q$7 (where the union of $Q$8 disjoint isomorphic copies of each $Q$9 is sufficient). In the trivial case $\Gamma_n$0, $\Gamma_n$1 itself is necessarily a minimal witness; for $\Gamma_n$2, any minimal witness has size exactly $\Gamma_n$3.

An immediate naïve expectation arises for general $\Gamma_n$4 that a tight bound could take the form $\Gamma_n$5, by chaining $\Gamma_n$6 posets in a path through “maximal element swaps.” However, the paper provides concrete counterexamples to this expectation, as spurious downsets can arise, requiring longer paths to circumvent undesired intersections and thus yielding witnesses with greater sizes.

Constructing Families With Arbitrarily Large Gaps

The authors construct explicit infinite families of examples to show that no universal bound of the form $\Gamma_n$7 (for any constant $\Gamma_n$8) can hold for the size of minimal witnesses. This is accomplished via three types of families:

1. Newtonian Posets: These highly regular posets, whose levels are completely connected downward, permit unions (type $\Gamma_n$9 posets) that exhibit the property $k$0 as the size increases, with $k$1 the minimal witness.
2. Tall Families With Arbitrary Seeds: By building posets out of chains topped with arbitrary seeds ($k$2, $k$3), unioned to prevent overlaps yielding connected substructures, the gap between the size of the minimal witness and $k$4 can be made unbounded as well.
3. Height 2 Families: Even restricting to posets of height 2 does not salvage a bound of the form $k$5: the constructed families show gaps growing linearly with the size parameter.

Conclusion: It is impossible to bound the size of the minimal witness for nodes of covtree by any linear expression $k$6 with constant $k$7.

The Exchange Graph and Improved General Bounds

To obtain improved, though non-tight, upper bounds valid in all circumstances, the paper introduces the exchange graph of downsets: a graph $k$8 whose vertices are downsets of size $k$9 of $n$0, with edges joining pairs differing by a single element.

The structural insight provided by this object is significant: $n$1 is always connected, and the distance between any two downsets is exactly the size of their symmetric difference (see Proposition on connectivity). Path-based analysis in $n$2 allows more refined counting of witness size, as illustrated in the following examples:

Figure 3: Exchange graph for a poset with a given labelling; this example demonstrates the necessity of special handling for very short paths.

Figure 4: The exchange graph $n$3 for a specific poset, with the minimal $n$4-$n$5-...-$n$6-$n$7 path shown bolded; this structure is crucial for precise witness size bounds.

The resulting general bound is that any minimal witness $n$8 for $n$9 with $n$0 must satisfy $n$1. For $n$2 the bound $n$3 is sharp. The improved general bound is obtained by utilizing the connectedness of the exchange graph, and constructing the witness as the union over the minimal paths between downsets.

Special Case: Tight Bounds for $n$4

Through an in-depth combinatorial analysis of exchange graph structure for triple-node sets, the authors prove a sharper bound for $n$5: If $n$6 consists of three elements, and a witness exists, then there is a minimal witness $n$7 with $n$8 The proof proceeds by analyzing shortest paths in the exchange graph between anchor copies, deriving structural constraints on potential downset intersections, and excluding pathological structures via careful combinatorial counting and subposet symmetries. Notably, further computational evidence in the small-$n$9 regime supports the conjecture that witnesses of size $\Gamma_n$0 suffice in all $\Gamma_n$1 cases, though a proof is as of yet lacking.

Implications and Future Directions

The findings impose precise limits on algorithmic approaches for explicit covtree construction: for $\Gamma_n$2, brute-force search may be halted at $\Gamma_n$3, yielding substantial efficiency savings. For higher $\Gamma_n$4, bounds are weaker but still informative.

The work also highlights combinatorial barriers in encoding label-independent poset growth and suggests the necessity of novel techniques to analyze the structure of witness sets and their embedding posets. The structural understanding of exchange graphs opens the possibility of “bootstrapping” covtree generation by combinatorial or algebraic invariants, potentially bypassing the need for brute-force search.

On the theoretical side, these bounds have an interrelation with causal set models of spacetime, where they inform the feasibility and uniqueness of reconstructing macroscopic histories from local downset data, and probe the interplay between growth and covariance in discrete quantum gravity models.

Conclusion

This work establishes, via explicit counterexamples and tight combinatorial reasoning, the nonexistence of linear bounds of the form $\Gamma_n$5 for minimal witness size in covtree, while also obtaining the best possible bounds in the $\Gamma_n$6 case and improved general bounds for larger $\Gamma_n$7 by exchange graph techniques. The results expose deep combinatorial obstacles to understanding the “growth” of unlabelled posets, with significant consequences for both physics (causal set theory) and pure combinatorics. They motivate new inquiries into the fine structure of witness posets, algorithmic enumeration, and potential bootstrapping methodologies within the covtree framework.