Quasi-Paving Matroids

• Quasi-paving matroids are generalized forms of paving matroids with circuits almost as large as the rank, bridging uniform and advanced matroid classes.
• They feature a combinatorial structure described by (k, ℓ)-uniform properties and hypergraph characterizations, facilitating precise minor control and algorithmic studies.
• Their robust closure properties, including duality and minor-closure, enable effective classification, optimization applications, and insights into tropical geometry.

A quasi-paving matroid is a matroid whose circuits (minimal dependent sets) are "almost" as large as the rank—specifically, one that relaxes the strict conditions on circuit sizes characteristic of paving matroids, but in a controlled manner. These objects are situated between classical paving matroids and more general matroidal families, and play a key role in structural, representability, and extremal questions in matroid theory. While the terminology for quasi-paving matroids is not completely standardized, across the literature, these often refer to classes such as (k, ℓ)-uniform matroids for parameters beyond the paving case, split matroids lying between paving and all matroids, or matroids with most, but not all, hyperplanes having prescribed independence or nullity properties.

1. Definitional Framework and Hierarchies

The prototypical paving matroid of rank $r$ is defined by the property that every circuit has size at least $r$—equivalently, every hyperplane is independent. The quasi-paving regime generalizes this strictness. Central to this generalization is the notion of $(k,\,\ell)$-uniform matroids: a matroid $M$ is $(k,\,\ell)$-uniform if it has no minor isomorphic to $U_{k,k}\oplus U_{0,\ell}$; equivalently, every rank-$(r(M)-k)$ flat $F$ of $M$ has nullity $|F|-r(F)<\ell$ (Drummond, 2021). The case $(k,\ell)=(2,1)$ recovers paving matroids, while increasing $k$ or $\ell$ allows more flexibility—matroids with "quasi-paving" properties in which some flats are allowed slightly higher nullity.

The following table highlights key thresholds:

Class	Defining property	Nullity/Forbidden minor characterization
Uniform matroid	All bases have the same size	$(1,1)$-uniform
Paving matroid	All circuits have size at least $r$	$(2,1)$-uniform ($U_{2,2}\oplus U_{0,1}$ forbidden)
Quasi-paving matroid (general)	Most hyperplanes independent; bounded nullity in coflats; circuits near size $r$	$(k,\ell)$-uniform for $k>2$ and/or $\ell>1$
Split/elementary split matroid	Hypergraph representation with hyperedges bounded locally; closed under minors/duality/trunc	Forbidden $U_{0,1}\oplus U_{1,2}\oplus U_{1,1}$ (Bérczi et al., 2022)

This gradation creates a hierarchy interpolating between uniformity, paving, and more general matroid behavior. The $(k,\ell)$-uniform property is particularly useful, as it generalizes paving and offers a parameterized approach for quantifying "distance" from uniformity.

2. Hypergraph and Polyhedral Characterizations

Quasi-paving matroids admit natural combinatorial and polyhedral descriptions that refine the familiar paving structure. Elementary split matroids, which generalize paving matroids, are characterized via hypergraph systems with intersection restrictions. Given a ground set $S$, a collection $\mathcal{H} = \{H_1, \ldots, H_q\}$, and parameters $r, r_1, \ldots, r_q$, the independent sets are:

$$\mathcal{I} = \{ X \subseteq S : |X| \leq r \;\text{and}\; |X \cap H_i| \leq r_i \text{ for all } i \}$$

with the consistency condition $|H_i \cap H_j| \leq r_i + r_j - r$ for all $i < j$ (Bérczi et al., 2022). The paving case recovers $r_i = r-1$ for all $i$, while quasi-paving/split classes allow $r_i < r-1$ for some (but not all) hyperedges.

A key distinction is that split (and in particular elementary split) matroids are closed under duality, taking minors, and truncation—properties not possessed by the paving class (Bérczi et al., 2022).

In polyhedral terms, matroid base polytopes can be "split" along certain hyperplanes, and the combinatorics of these splits are reflected in the structure of the matroid (Bérczi et al., 2022). This approach is closely tied to tropical and Dressian geometry.

3. Examples, Extremal Objects, and Binary Quasi-Paving Matroids

An explicit structural classification exists for certain subclasses, especially over GF(2). For binary $(2,2)$-uniform matroids, the maximally 3-connected members are:

• $Z_5 \setminus t$, the tipless binary 5-spike,
• $AG(4,2)$ and $AG(4,2)^*$ (the affine geometry and its dual over GF(2)),
• The self-dual matroid $P_{10}$ (Drummond, 2021).

All these objects have controlled deviation from the uniform or paving regime, manifesting the quasi-paving condition through bounded nullity on large flats.

Moreover, split and elementary split matroids admit forbidden minor characterizations. In particular, elementary split matroids are characterized precisely as those having no $U_{0,1}\oplus U_{1,2}\oplus U_{1,1}$-minor (Bérczi et al., 2022). Their closed lists in the binary setting include strong paving analogues, such as connected binary (sparse) paving matroids of higher rank.

4. Structural and Representability Implications

An important feature of $(k,\ell)$-uniform and split/quasi-paving matroids is the existence of "finiteness" and minor control in representable classes. Specifically, for any $(k,\ell)$ and finite $q$, there are only finitely many simple, cosimple $GF(q)$-representable $(k,\ell)$-uniform matroids (Drummond, 2021). Assuming Rota's Conjecture, for every finite field there are parameters $(k_q, \ell_q)$ such that all excluded minors for $GF(q)$-representability are $(k_q, \ell_q)$-uniform.

This situates quasi-paving matroids as natural candidates for understanding and bounding excluded minors—key objects in matroid structural theory and cryptomorphisms.

Additionally, split and elementary split matroids offer improved algorithmic accessibility due to the clarity of their hypergraph definitions and closure properties, facilitating applications in combinatorial optimization and tropical geometry (Bérczi et al., 2022).

5. Connections to Paving Matroids, Excluded Minors, and Tropical Ideals

Quasi-paving matroids are closely tied to the general paving conjecture, which posits that almost all matroids are paving as $n\to\infty$. Split and quasi-paving matroids, by including paving as a subclass and improving closure properties, serve as plausible "typical" matroids in large-rank or large-ground-set limits (Bérczi et al., 2022).

Classification results for excluded minors of minor-closed matroid classes often employ quasi-paving matroids. Because $(k,\ell)$-uniform matroids have bounded rank when representable over a fixed field (Drummond, 2021), their appearance in excluded minor lists is inherent and provides a means to control the wildness of forbidden structures.

In tropical geometry, similar patterns appear: for example, paving tropical ideals correspond to matroids whose circuits are all rank or rank$+$1 in size, and a plausible extension is that quasi-paving tropical ideals admit circuits in a slightly broader size range (Anderson et al., 2021). This suggests a framework for moduli spaces of tropical schemes, parameterized by quasi-paving data such as invariant d-partitions or hypergraph clutters.

6. Applications, Algorithmic Advantages, and Future Directions

Quasi-paving matroids, particularly those realized via hypergraph systems or in the split class, inherit much of the favorable combinatorial structure of pavings. Their closure under duals, minors, and truncation makes them robust in optimization and geometric contexts. Their role in bounding representability (through $(k,\ell)$-uniformity), structural analysis of excluded minors, and combinatorial models for tropicalization indicates broad applicability (Drummond, 2021, Bérczi et al., 2022, Anderson et al., 2021).

A plausible implication is that quasi-paving matroids, being both general and amenable to explicit combinatorial control, will become central to ongoing classification efforts in matroid theory, especially in unexplored or computationally challenging settings.

• "A generalisation of uniform matroids" (Drummond, 2021)
• "Hypergraph characterization of split matroids" (Bérczi et al., 2022)
• "Paving Tropical Ideals" (Anderson et al., 2021)