Crossing Families in Combinatorics

• Crossing family is a multifaceted concept defined variably across settings: in planar point sets, geometric graphs, set systems, and lattice theory.
• Key results include linear lower bounds in planar geometry and tight extremal estimates in graph and set-theoretic formulations.
• Methodologies such as uncrossing, Ramsey-type arguments, and submodular flows underpin constructive bounds and algorithmic advances.

In current combinatorics and graph theory, the term crossing family is used for several non-equivalent structures that share a common organizing idea: crossing is treated as a forbidden, enforced, or closure-generating relation. In planar point-set geometry, a crossing family is a collection of segments whose interiors pairwise intersect; in geometric-graph generalizations, the objects can be vertex-disjoint copies of a fixed graph $H$; in combinatorial optimization, a crossing family is a set system closed under intersection and union of crossing sets; and in extremal set theory, two subsets are crossing when all four regions of their Venn diagram are nonempty. These notions are linked by uncrossing arguments, separator methods, Ramsey-type extraction, and extremal bounds, but they arise in distinct technical settings (Lara et al., 2018, Abdi et al., 2024, Tomon, 5 Mar 2026).

1. Terminology and core definitions

The literature represented here uses the same phrase for several different objects.

Context	Definition	Representative source
Planar point sets	A collection of segments spanned by a point set such that every two segments intersect internally	(Aichholzer et al., 2021)
Geometric graphs	A set of pairwise crossing, vertex-disjoint copies of a fixed graph $H$	(Lara et al., 2018)
Set systems on $V$	A family $\mathcal{C}\subseteq 2^V$ closed under $U\cap W$ and $U\cup W$ whenever $U,W\in\mathcal{C}$ cross	(Abdi et al., 2024)
Extremal set theory	Two subsets $A,B\subset X$ are crossing if $A\setminus B$, $B\setminus A$, $H$0, and $H$1 are all nonempty	(Tomon, 5 Mar 2026)
Integer lattices	Two vectors $H$2 are $H$3-crossing if $H$4 and $H$5 for some coordinates $H$6	(Lasoń et al., 2012)

For planar point sets $H$7, the standard notation is

$H$8

and

$H$9

A $V$0-crossing family is then a set of $V$1 segments spanned by $V$2 such that all $V$3 segments mutually cross in their interiors (Aichholzer et al., 2021).

In the generalized geometric-graph setting, if $V$4 is a fixed graph, an $V$5-crossing family in a geometric graph $V$6 is a set of pairwise crossing, vertex-disjoint copies of $V$7 in $V$8. For $V$9, this recovers the original notion (Lara et al., 2018).

In the optimization-oriented set-system setting, a crossing family $\mathcal{C}\subseteq 2^V$0 over a finite ground set $\mathcal{C}\subseteq 2^V$1 satisfies the closure rule that if $\mathcal{C}\subseteq 2^V$2 have $\mathcal{C}\subseteq 2^V$3 and $\mathcal{C}\subseteq 2^V$4, then both $\mathcal{C}\subseteq 2^V$5 and $\mathcal{C}\subseteq 2^V$6 also belong to $\mathcal{C}\subseteq 2^V$7 (Abdi et al., 2024). This is the uncrossing framework used in orientation and dijoin problems.

2. Geometric crossing families of segments in planar point sets

The geometric notion originates in work cited by later papers as having been introduced by Aronov et al. In that framework, every complete geometric graph on $\mathcal{C}\subseteq 2^V$8 points in general position contains a $\mathcal{C}\subseteq 2^V$9-crossing family of size at least $U\cap W$0 (Lara et al., 2018). At the opposite extreme, point sets exist whose maximum crossing family uses substantially fewer than $U\cap W$1 points.

The upper-bound side has improved incrementally. A 2019 construction showed that for all $U\cap W$2, there exist $U\cap W$3-point sets whose crossing family has size at most

$U\cap W$4

improving the earlier $U\cap W$5 upper bound (Evans et al., 2019). A later computational and constructive improvement produced point sets with no crossing family larger than

$U\cap W$6

that is,

$U\cap W$7

The same work determined the exact threshold for $U\cap W$8-crossing families on small point sets: every set of at least $U\cap W$9 points in general position contains a $U\cup W$0-crossing family, $U\cup W$1 for $U\cup W$2, and $U\cup W$3 for $U\cup W$4 (Aichholzer et al., 2021).

The 2025 Ramsey-type strengthening relates crossing families to a distinct notion called a non-crossing family of size $U\cup W$5, consisting of four disjoint subsets $U\cup W$6, each of size $U\cup W$7, such that for every choice of representatives $U\cup W$8, the point $U\cup W$9 lies in the interior of the triangle formed by $U,W\in\mathcal{C}$0. The main theorem states that for every $U,W\in\mathcal{C}$1, every set $U,W\in\mathcal{C}$2 of $U,W\in\mathcal{C}$3 points in the plane in general position contains either a crossing family of size

$U,W\in\mathcal{C}$4

or a non-crossing family of size $U,W\in\mathcal{C}$5. The proof is constructive, with expected time $U,W\in\mathcal{C}$6, and the paper also proves that a crossing family of size $U,W\in\mathcal{C}$7 or a non-crossing family of size $U,W\in\mathcal{C}$8 can be found in expected time $U,W\in\mathcal{C}$9 (Antić et al., 24 Aug 2025). For fixed $A,B\subset X$0, this implies a linear lower bound.

These results leave intact the central open problem emphasized across the literature: whether the largest crossing family guaranteed in every $A,B\subset X$1-point set is linear in $A,B\subset X$2 (Evans et al., 2019, Antić et al., 24 Aug 2025).

3. Generalizations to geometric graphs and intersecting configurations

The notion of crossing family extends naturally from edges to larger geometric subgraphs. For a fixed graph $A,B\subset X$3, an $A,B\subset X$4-crossing family consists of pairwise crossing, vertex-disjoint copies of $A,B\subset X$5 in a complete geometric graph $A,B\subset X$6 (Lara et al., 2018). This generalization changes the extremal behavior substantially.

For paths, stars, and cliques, explicit lower bounds are known. Every $A,B\subset X$7 contains a $A,B\subset X$8-crossing family of size at least

$A,B\subset X$9

a $A\setminus B$0-crossing family of size at least

$A\setminus B$1

and, more generally, a $A\setminus B$2-crossing family of size $A\setminus B$3 for $A\setminus B$4. For complete graphs, every $A\setminus B$5 contains a $A\setminus B$6-crossing family of size at least

$A\setminus B$7

and for any $A\setminus B$8, a $A\setminus B$9-crossing family of size at least $B\setminus A$0 (Lara et al., 2018). The same paper records exact small-threshold values $B\setminus A$1 and $B\setminus A$2, where $B\setminus A$3 is the minimum $B\setminus A$4 such that every complete geometric graph on $B\setminus A$5 vertices has an $B\setminus A$6-crossing family of size $B\setminus A$7.

A parallel notion is the intersecting family, in which the subgraphs are required to be edge-disjoint rather than vertex-disjoint. In this sense, any set $B\setminus A$8 of $B\setminus A$9 points in general position admits an intersecting family of edge-disjoint triangles of size at least

$H$00

confirming a conjecture of Lara and Rubio-Montiel (Álvarez-Rebollar et al., 2022).

Several additional generalized crossing structures have been analyzed. An elbow is a chain of one horizontal and one vertical segment connecting two points of $H$01, and every point set of size $H$02 contains a mutually crossing family of elbows of size at least $H$03, while some point sets admit no more than $H$04 mutually crossing elbows (Álvarez-Rebollar et al., 2022). From any family of $H$05 mutually crossing triangles, one can always extract at least $H$06 mutually crossing $H$07-paths by deleting one edge from each triangle, and there is an example showing that $H$08 cannot be taken larger than $H$09 (Álvarez-Rebollar et al., 2022). For every $H$10, there is a constant $H$11 such that any sufficiently large point set admits a mutually crossing family of at least $H$12 simple convex $H$13-cycles; for $H$14, one may take $H$15 (Álvarez-Rebollar et al., 2022).

A different but related formalism is the $H$16-crossing family in a geometric graph: a pair of edge subsets $H$17 with $H$18, $H$19, each set pairwise crossing internally, and every edge in $H$20 disjoint from every edge in $H$21 (Fulek et al., 2010). The conjectured extremal behavior is linear: for fixed $H$22, every $H$23-vertex geometric graph with no $H$24-crossing family should have at most $H$25 edges. The best bounds cited here are that forbidding a $H$26-crossing family yields at most $H$27 edges, forbidding a $H$28-crossing family yields $H$29 edges, and in simple topological graphs, forbidding a $H$30-crossing family yields at most $H$31 edges (Fulek et al., 2010).

4. Crossing families as crossing-closed set systems

In combinatorial optimization and orientation theory, a crossing family is not a family of geometric objects but a family of subsets closed under uncrossing operations. If $H$32 is a crossing family over $H$33, and $H$34 is a connected graph, then a strong orientation for $H$35 is an orientation of $H$36 such that each $H$37 has at least one outgoing and at least one incoming arc (Abdi et al., 2024).

The main theorem in this setting is exact: if $H$38 is connected and $H$39 is a crossing family over $H$40 such that $H$41 for every $H$42, then there exists a strong orientation of $H$43 for $H$44 (Abdi et al., 2024). The proof is polyhedral. It reformulates the problem as a $H$45 feasibility system on arc reversals, interprets the constraints as the intersection of two submodular-flow systems, and uses total dual integrality. The same paper states that this implies the main conjecture in Chudnovsky et al. (2016), and in particular that in every minimal counterexample to the Edmonds-Giles conjecture where the minimum weight of a dicut is $H$46, the arcs of nonzero weight must be disconnected (Abdi et al., 2024).

A 2026 extension studies cosignings of crossing families. A signing $H$47 is a cosigning if every set in $H$48 includes a positive element and excludes a negative element; it is $H$49-closed if every pairwise nonempty intersection and co-intersection include positive and negative elements, respectively (Abdi et al., 27 Feb 2026). The paper gives necessary and sufficient conditions for the existence of both ordinary and $H$50-closed cosignings, proves polynomial-time forcing algorithms, and further shows that the cosigning algorithm can be run in oracle polynomial time via submodular function minimization (Abdi et al., 27 Feb 2026).

Cosigned crossing families arise naturally in digraphs whose vertex set is split into sources and sinks, with every set in $H$51 covered by an incoming arc (Abdi et al., 27 Feb 2026). Under mild and necessary conditions, the same paper constructs an outer-planar arc covering of $H$52 when the vertices are placed around a circle. These outer-planar gadgets are then used to find disjoint dijoins in $H$53-weighted planar digraphs when the weight-$H$54 arcs form a connected component that is not necessarily spanning (Abdi et al., 27 Feb 2026).

5. Set-theoretic and lattice-theoretic variants

A distinct usage in extremal set theory defines crossing directly on pairs of subsets. If $H$55, they are crossing when none of the sets

$H$56

is empty (Tomon, 5 Mar 2026). The central question here is not the extraction of a crossing family, but the size of a family that avoids many pairwise crossing members.

Karzanov and Lomonosov conjectured that every family $H$57, with $H$58, that does not contain $H$59 pairwise crossing members has size $H$60. This has now been resolved in its growth-rate form: for every $H$61, there exists $H$62 such that every $H$63-cross-free family $H$64 with $H$65 satisfies

$H$66

Thus $H$67 (Tomon, 5 Mar 2026). The cited historical progression is explicit: for $H$68, the maximum size is $H$69; for $H$70, optimal linear bounds such as $H$71 were known; for $H$72, the best previous upper bounds were $H$73 and later $H$74 (Tomon, 5 Mar 2026). The proof uses a weakened notion of crossing, Dilworth’s theorem, Turán’s theorem, chain extraction, and cross-support trees (Tomon, 5 Mar 2026). The paper also records applications to network flow theory, discrete geometry, and phylogenetics, and notes that in convex position geometric and set-theoretic crossing coincide (Tomon, 5 Mar 2026).

Another lattice-theoretic variant appears in the study of crossing vectors. Two vectors $H$75 are $H$76-crossing if there exist coordinates $H$77 such that

$H$78

The extremal function $H$79 is the maximum size of a family of pairwise $H$80-crossing and pairwise non-$H$81-crossing vectors in $H$82 (Lasoń et al., 2012). The conjecture is

$H$83

This is proved for $H$84, while for $H$85 the known bounds are

$H$86

(Lasoń et al., 2012). The motivation comes from the width of the lattice of maximum antichains of a partially ordered set (Lasoń et al., 2012).

6. Methods, applications, and open directions

The theory of crossing families is methodologically heterogeneous. In geometric settings, classical tools include mutually avoiding sets and the Erdős-Szekeres theorem (Lara et al., 2018), partitioning arguments based on Ceder’s theorem (Lara et al., 2018), exhaustive generation of order types together with SAT verification (Aichholzer et al., 2021), semi-algebraic Ramsey methods (Álvarez-Rebollar et al., 2022), and same-type lemmas plus regularity lemmas for semi-algebraic hypergraphs (Antić et al., 24 Aug 2025). These techniques are used either to force pairwise crossing substructures or to build point configurations that suppress them.

In set-system and optimization settings, the dominant techniques are uncrossing, chain decompositions, submodular flows, total dual integrality, and submodular function minimization (Abdi et al., 2024, Abdi et al., 27 Feb 2026, Tomon, 5 Mar 2026). The common pattern is that crossing creates enough closure or rigidity to permit either compact structure theorems or polynomial-time algorithms.

Several open problems remain explicit in the cited literature. The principal geometric question is whether every $H$87-point set in general position contains a crossing family of size $H$88 (Evans et al., 2019, Antić et al., 24 Aug 2025). For elbows, it is conjectured that the universal lower bound $H$89 is tight (Álvarez-Rebollar et al., 2022). For generalized intersecting families, the existence of quadratic-size $H$90-intersecting and $H$91-intersecting families in $H$92 is posed as conjectural (Lara et al., 2018). In the optimization variant, polynomial-time oracle algorithms are available for ordinary cosigning, but not for the $H$93-closed version (Abdi et al., 27 Feb 2026). In the vector setting, the formula $H$94 remains open for $H$95 (Lasoń et al., 2012).

Taken together, these results show that crossing family is less a single object than a cluster of related concepts. The unifying theme is that crossing, whether interpreted geometrically, set-theoretically, or order-theoretically, imposes enough combinatorial tension to support sharp extremal estimates, structural closure theorems, and constructive algorithms.