Non-Crossing Family in Planar Geometry

• Non-crossing family is defined as four disjoint subsets of a planar point set such that every transversal yields three points forming a triangle with the fourth lying strictly inside it.
• The concept provides a structured alternative to crossing families, establishing a dichotomy that enables linear and subexponential-time algorithms in geometric Ramsey theory.
• It plays a pivotal role in extremal combinatorics by linking convex bundles, cup/cap patterns, and bipartite methods to guarantee either large crossing or non-crossing families.

Searching arXiv for the most relevant papers on “non-crossing family,” with emphasis on exact terminology and nearby uses. Searching arXiv for the exact phrase and closely related formulations. A non-crossing family, in the planar point-set sense introduced in "Crossing and non-crossing families," is a collection of four pairwise disjoint subsets $P_1,P_2,P_3,P_4$ of a finite point set $P\subset \mathbb{R}^2$ in general position such that for every transversal choice $p_i\in P_i$, the 4-point set $\{p_1,p_2,p_3,p_4\}$ is not in convex position, with $p_4$ lying in the interior of the triangle spanned by $p_1,p_2,p_3$. The size of such a family is $\min\{|P_1|,|P_2|,|P_3|,|P_4|\}$. In this form, the notion was introduced as a structured alternative to a crossing family and as a parameter governing stronger lower bounds for crossing families in planar point sets (Antić et al., 24 Aug 2025).

1. Definition and geometric content

The ambient setting is a finite point set $P\subset \mathbb{R}^2$ in general position, meaning that no three points of $P$ are collinear. A non-crossing family is then a collection of four pairwise disjoint non-empty subsets

$P_1,P_2,P_3,P_4\subseteq P$

such that for every choice of points $P\subset \mathbb{R}^2$0, the point $P\subset \mathbb{R}^2$1 lies in the interior of the triangle formed by $P\subset \mathbb{R}^2$2. The paper also defines the size of a non-crossing family as

$P\subset \mathbb{R}^2$3

Thus, saying that $P\subset \mathbb{R}^2$4 contains a non-crossing family of size $P\subset \mathbb{R}^2$5 means that there exist four disjoint subsets each of cardinality at least $P\subset \mathbb{R}^2$6, and in the theorem statements they are taken to have cardinality exactly $P\subset \mathbb{R}^2$7 (Antić et al., 24 Aug 2025).

Geometrically, the four parts are constrained so rigidly that every transversal of one point from each part has the same convexity pattern: the first three points form a triangle and the fourth point always lies strictly inside that triangle. This suggests that the notion is best understood as a robust transversal witness of non-convexity rather than merely as the existence of one exceptional 4-tuple. The paper explicitly notes that, by Carathéodory’s theorem, a point set is in convex position if and only if it does not contain a non-crossing family of size $P\subset \mathbb{R}^2$8 (Antić et al., 24 Aug 2025).

The terminology is motivated by contrast with a crossing family, which is a collection of pairwise crossing segments. Here the role of the non-crossing family is not to encode pairwise nonintersection of segments, but rather to package a uniform non-convexity relation across four large classes of points. The name is therefore contextual: it refers to the structured alternative to large crossing families in planar Ramsey-type statements (Antić et al., 24 Aug 2025).

2. Dichotomy with crossing families

The central theorems place non-crossing families opposite crossing families. The first main result asserts the existence of a constant $P\subset \mathbb{R}^2$9 such that, for all positive integers $p_i\in P_i$0 and $p_i\in P_i$1, every set of at least $p_i\in P_i$2 points in the plane in general position contains either a convex bundle of size $p_i\in P_i$3 and width $p_i\in P_i$4, or a non-crossing family of size $p_i\in P_i$5. Since a convex bundle of size $p_i\in P_i$6 yields a crossing family of size $p_i\in P_i$7, the paper derives the corollary that every set of at least $p_i\in P_i$8 points contains either a crossing family of size $p_i\in P_i$9 or a non-crossing family of size $\{p_1,p_2,p_3,p_4\}$0 (Antić et al., 24 Aug 2025).

This already gives the linear tradeoff

$\{p_1,p_2,p_3,p_4\}$1

unless a non-crossing family of size $\{p_1,p_2,p_3,p_4\}$2 is present. The stronger theorem replaces this linear-in-$\{p_1,p_2,p_3,p_4\}$3 tradeoff by a subexponential denominator in $\{p_1,p_2,p_3,p_4\}$4: there is a constant $\{p_1,p_2,p_3,p_4\}$5 such that, for every positive integer $\{p_1,p_2,p_3,p_4\}$6, every set of $\{p_1,p_2,p_3,p_4\}$7 points in the plane in general position contains either a non-crossing family of size $\{p_1,p_2,p_3,p_4\}$8, or a crossing family of size

$\{p_1,p_2,p_3,p_4\}$9

Setting $p_4$0 recovers the Pach–Rubin–Tardos lower bound

$p_4$1

so the theorem is a genuine strengthening: when the largest forbidden non-crossing family is substantially smaller than $p_4$2, the guaranteed crossing family becomes correspondingly larger (Antić et al., 24 Aug 2025).

A further technical ingredient is a bipartite version of the Pach–Rubin–Tardos theorem. If $p_4$3 is partitioned into two separated subsets $p_4$4 with $p_4$5, then $p_4$6 contains a crossing family of size at least

$p_4$7

all of whose segments have one endpoint in $p_4$8 and the other in $p_4$9. This theorem is used inside the proof of the main dichotomy by applying it to opposite classes of a convex bundle (Antić et al., 24 Aug 2025).

3. Constructive and algorithmic aspects

The paper is constructive. It proves that there is a constant $p_1,p_2,p_3$0 such that for all positive integers $p_1,p_2,p_3$1, if $p_1,p_2,p_3$2 is a set of $p_1,p_2,p_3$3 points in the plane in general position, then a convex bundle of size $p_1,p_2,p_3$4 and width $p_1,p_2,p_3$5 or a non-crossing family of size $p_1,p_2,p_3$6 can be computed in expected time

$p_1,p_2,p_3$7

Combining this with the implication from convex bundles to crossing families yields an expected linear-time algorithm that finds either a crossing family of size $p_1,p_2,p_3$8 or a non-crossing family of size $p_1,p_2,p_3$9 (Antić et al., 24 Aug 2025).

The stronger algorithmic theorem states that a crossing family of size

$\min\{|P_1|,|P_2|,|P_3|,|P_4|\}$0

or a non-crossing family of size $\min\{|P_1|,|P_2|,|P_3|,|P_4|\}$1 can be computed in expected time

$\min\{|P_1|,|P_2|,|P_3|,|P_4|\}$2

The abstract records the same bound in the form $\min\{|P_1|,|P_2|,|P_3|,|P_4|\}$3. The runtime arises by first computing a convex bundle or non-crossing family, and then, in the convex-bundle case, running the Pach–Rubin–Tardos algorithm independently on pairs of opposite classes of size $\min\{|P_1|,|P_2|,|P_3|,|P_4|\}$4 (Antić et al., 24 Aug 2025).

Algorithmically, the decisive tool is the Same-Type Lemma together with Rubin’s semi-algebraic regularity lemma. The paper analyzes the constructive proof of the Same-Type Lemma and states the resulting time bound

$\min\{|P_1|,|P_2|,|P_3|,|P_4|\}$5

for extracting same-type subsets from $\min\{|P_1|,|P_2|,|P_3|,|P_4|\}$6 disjoint point sets in $\min\{|P_1|,|P_2|,|P_3|,|P_4|\}$7. Because the arguments in the convex-bundle theorem only use constant values $\min\{|P_1|,|P_2|,|P_3|,|P_4|\}$8 and $\min\{|P_1|,|P_2|,|P_3|,|P_4|\}$9, these extractions are linear in the relevant input size, which is what makes the $P\subset \mathbb{R}^2$0 expected-time bound possible (Antić et al., 24 Aug 2025).

4. Proof architecture

The proof of the convex-bundle theorem begins by partitioning the point set into seven vertical slabs

$P\subset \mathbb{R}^2$1

each of equal size. The Same-Type Lemma is then applied to obtain subsets

$P\subset \mathbb{R}^2$2

such that every 7-transversal has the same order type. If one such transversal were not in convex position, then by Carathéodory’s theorem some 4-subset would already be non-convex; the same-type property would then force the corresponding four classes to form a non-crossing family of size $P\subset \mathbb{R}^2$3, contradicting the hypothesis. Hence every such transversal is in convex position (Antić et al., 24 Aug 2025).

From there, the proof organizes the transversals into a cup/cap pattern. There exist index sets $P\subset \mathbb{R}^2$4 with

$P\subset \mathbb{R}^2$5

such that the points indexed by $P\subset \mathbb{R}^2$6 form a cap and those indexed by $P\subset \mathbb{R}^2$7 form a cup. By pigeonhole, one of these has size at least $P\subset \mathbb{R}^2$8. The proof then iteratively builds many classes

$P\subset \mathbb{R}^2$9

and subsets

$P$0

so that every transversal forms an $P$1-cap, where $P$2 is the zig-zag permutation

$P$3

At each step, a same-type extraction is used to rule out the “wrong” placement of the next class: such a placement would again force a non-crossing family of size at least $P$4 (Antić et al., 24 Aug 2025).

Once a convex bundle

$P$5

has been obtained, the bipartite Pach–Rubin–Tardos theorem is applied independently to each $P$6. Each pair yields a crossing family $P$7 of size at least

$P$8

for some absolute constant $P$9. Because every segment joining $P_1,P_2,P_3,P_4\subseteq P$0 to $P_1,P_2,P_3,P_4\subseteq P$1 crosses every segment joining $P_1,P_2,P_3,P_4\subseteq P$2 to $P_1,P_2,P_3,P_4\subseteq P$3 for $P_1,P_2,P_3,P_4\subseteq P$4, the union

$P_1,P_2,P_3,P_4\subseteq P$5

is itself a crossing family, of total size

$P_1,P_2,P_3,P_4\subseteq P$6

This is the point where the absence of a large non-crossing family translates into the presence of a large crossing family (Antić et al., 24 Aug 2025).

5. Terminological variants in adjacent literatures

The phrase non-crossing family is not uniform across combinatorics and geometry. In some papers it is the exact primary term; in others the closest notion is a family of pairwise noncrossing objects, or a class defined by avoiding crossing patterns.

Context	Meaning of “non-crossing”	Source
Planar point sets	Four disjoint classes $P_1,P_2,P_3,P_4\subseteq P$7 whose every transversal has $P_1,P_2,P_3,P_4\subseteq P$8 inside $P_1,P_2,P_3,P_4\subseteq P$9	(Antić et al., 24 Aug 2025)
Multiset permutations	Permutations of $P\subset \mathbb{R}^2$00 avoiding $P\subset \mathbb{R}^2$01 and $P\subset \mathbb{R}^2$02; equivalently non-crossing matchings	(Archer et al., 18 Feb 2025)
$P\subset \mathbb{R}^2$03-subsets of $P\subset \mathbb{R}^2$04	A face of $P\subset \mathbb{R}^2$05, i.e. a pairwise noncrossing family of $P\subset \mathbb{R}^2$06-subsets	(Santos et al., 2014)
Planar partitions	Partitions whose block convex hulls are pairwise disjoint, giving $P\subset \mathbb{R}^2$07 or the classical lattice $P\subset \mathbb{R}^2$08	(Cohen et al., 2023, Baumeister et al., 2019)
Geometric graphs	No exact “non-crossing family” term; closest is disjointness between two internally pairwise crossing groups in a $P\subset \mathbb{R}^2$09-crossing family	(Fulek et al., 2010)

In permutation theory, the term refers to a class of permutations on $P\subset \mathbb{R}^2$10 that avoid the crossing patterns $P\subset \mathbb{R}^2$11. The paper "Pattern avoidance in non-crossing and non-nesting permutations" studies the subclass additionally avoiding $P\subset \mathbb{R}^2$12, obtains an algebraic quartic equation for the generating function, and notes symmetry with avoidance of $P\subset \mathbb{R}^2$13 (Archer et al., 18 Feb 2025).

In higher-dimensional Catalan combinatorics, "Noncrossing sets and a Graßmann associahedron" defines the flag complex $P\subset \mathbb{R}^2$14 on $P\subset \mathbb{R}^2$15, where a face is exactly a pairwise noncrossing family of $P\subset \mathbb{R}^2$16-subsets. The paper proves that $P\subset \mathbb{R}^2$17 is a flag, regular, unimodular and Gorenstein triangulation of the order polytope $P\subset \mathbb{R}^2$18, and that its reduced complex is a flag polytopal sphere dual to the Graßmann associahedron (Santos et al., 2014).

In the partition literature, noncrossing typically means that convex hulls of blocks are pairwise disjoint. "Noncrossing Partition Lattices from Planar Configurations" defines $P\subset \mathbb{R}^2$19 for a finite planar configuration $P\subset \mathbb{R}^2$20, while the survey "Non-crossing partitions" treats the classical cyclically ordered case $P\subset \mathbb{R}^2$21, its lattice structure, Kreweras complement, and Coxeter-theoretic generalizations (Cohen et al., 2023, Baumeister et al., 2019). This variation in usage indicates that the phrase denotes a family shaped by a crossing-avoidance relation, but the underlying objects—point classes, permutations, subsets, or partitions—depend strongly on the surrounding theory.

6. Broader combinatorial landscape and open direction

The current planar-point-set notion sits naturally beside a large body of work where noncrossing structures organize decomposition, enumeration, and duality. In positroid theory, every positroid decomposes uniquely according to a non-crossing partition of its cyclically ordered ground set, and the face poset of a positroid polytope embeds in a poset of weighted non-crossing partitions (Ardila et al., 2013). In random combinatorics, "Simply generated non-crossing partitions" studies weighted models on $P\subset \mathbb{R}^2$22 via a bijection to simply generated plane trees, where blocks of size $P\subset \mathbb{R}^2$23 correspond exactly to vertices of outdegree $P\subset \mathbb{R}^2$24 (Kortchemski et al., 2015). In higher Catalan geometry, "Noncrossing hypertrees" identifies the noncrossing hypertree complex with a generalized cluster complex of type $P\subset \mathbb{R}^2$25 and proves that it is naturally homeomorphic to the noncrossing partition link (McCammond, 2017). In lattice theory, "Noncommutative crossing partitions" enlarges the classical non-crossing partition lattice to a graded lattice containing the Kreweras lattice as a sublattice (Shigechi, 2022).

Against that backdrop, the planar-point-set non-crossing family of (Antić et al., 24 Aug 2025) is notable because it is not a family of pairwise noncrossing geometric primitives such as chords, blocks, or subsets. Instead it is a four-part transversal condition expressing a uniform interior-point relation. This suggests that its role is closer to that of a forbidden configuration in geometric Ramsey theory than to a Catalan object.

The main open problem emphasized in the planar-point-set paper asks whether there exists a constant $P\subset \mathbb{R}^2$26 and a function $P\subset \mathbb{R}^2$27 with $P\subset \mathbb{R}^2$28 such that every set of $P\subset \mathbb{R}^2$29 points in the plane in general position contains either a crossing family of size at least $P\subset \mathbb{R}^2$30 or a non-crossing family of size at least $P\subset \mathbb{R}^2$31. In the language of that paper, the known theorem

$P\subset \mathbb{R}^2$32

does not yet reach the linear-vs.-linear form suggested by Orthaber’s problem (Antić et al., 24 Aug 2025). This leaves the non-crossing family as both a concrete geometric notion and a parameter around which a broader extremal theory is still developing.