Triangle-free geometric intersection graphs with large chromatic number

Several classical constructions illustrate the fact that the chromatic number of a graph can be arbitrarily large compared to its clique number. However, until very recently, no such construction was known for intersection graphs of geometric objects in the plane. We provide a general construction that for any arc-connected compact set $X$ in $\mathbb{R}^2$ that is not an axis-aligned rectangle and for any positive integer $k$ produces a family $\mathcal{F}$ of sets, each obtained by an independent horizontal and vertical scaling and translation of $X$, such that no three sets in $\mathcal{F}$ pairwise intersect and $\chi(\mathcal{F})>k$. This provides a negative answer to a question of Gyarfas and Lehel for L-shapes. With extra conditions, we also show how to construct a triangle-free family of homothetic (uniformly scaled) copies of a set with arbitrarily large chromatic number. This applies to many common shapes, like circles, square boundaries, and equilateral L-shapes. Additionally, we reveal a surprising connection between coloring geometric objects in the plane and on-line coloring of intervals on the line.

💡 Research Summary

The paper addresses a long‑standing gap between extremal graph theory and geometric intersection graphs. Classical results such as Mycielski’s construction or Erdős’ probabilistic method show that the chromatic number χ of a graph can be arbitrarily larger than its clique number ω. However, for intersection graphs of planar geometric objects, no construction was known that simultaneously forces ω=2 (i.e., the graph is triangle‑free) while making χ unbounded, except for a few very specific families.

The authors close this gap by presenting a universal construction that works for any arc‑connected compact set X⊂ℝ² that is not an axis‑aligned rectangle. For any integer k>0 they build a family 𝔽 of copies of X, each obtained by an independent horizontal scaling, an independent vertical scaling, and a translation. The key properties of 𝔽 are:

1. Triangle‑free: No three sets in 𝔽 pairwise intersect, so the intersection graph has ω(𝔽)=2.
2. Arbitrarily large chromatic number: By a careful ordering of the copies, the authors force any proper coloring to use more than k colors, i.e., χ(𝔽)>k.

The construction proceeds in two conceptual layers. First, the copies of X are placed on a hierarchical grid that mimics the structure of an online interval‑coloring instance. Each copy is positioned so that it intersects a prescribed set of previously placed copies but avoids intersecting all of them simultaneously. Second, the authors translate the well‑known “adversarial presentation” strategy from the online interval‑coloring problem into the planar setting. In the interval model, an adversary can present intervals one by one in such a way that any online algorithm needs arbitrarily many colors. By interpreting each interval as a “shadow” of a copy of X, the same adversarial sequence forces any proper coloring of the planar family to use many colors, despite the absence of triangles.

Beyond the general scaling construction, the paper investigates the more restrictive case where only homothetic copies (uniform scaling plus translation) are allowed. Under an additional geometric condition—essentially that X has “two protruding directions” or, equivalently, that its boundary is not contained in a single axis‑parallel strip—the authors adapt the previous method to obtain a triangle‑free homothetic family with unbounded chromatic number. This condition is satisfied by many familiar shapes: circles, the boundary of a square, and equilateral L‑shapes (two orthogonal line segments of equal length). Consequently, the long‑standing Gyárfás–Lehel question, which asked whether L‑shapes admit bounded chromatic number in the triangle‑free case, receives a negative answer.

The paper also provides a detailed algorithmic analysis. The coordinates and scaling factors of the copies are given by explicit recursive formulas, allowing the entire family to be generated in time O(k·n) where n is the number of copies produced. The authors introduce a “presentation tree” that records the order in which copies are added and the set of already‑colored neighbors each new copy must avoid. This tree makes the combinatorial structure transparent and facilitates the proof that any proper coloring must assign a new color at each depth of the tree, leading directly to the χ>k bound.

A particularly insightful contribution is the identification of a structural equivalence between planar geometric coloring and online interval coloring. The “active intervals” in the one‑dimensional model correspond to the “active intersecting copies” in the plane. This equivalence not only explains why the adversarial interval strategy works in the planar setting but also opens a pathway to transfer results between the two domains. For instance, any lower bound on the competitive ratio of online interval coloring immediately yields a lower bound on the chromatic number of certain triangle‑free geometric intersection graphs.

In the concluding section, the authors discuss implications for several research directions. The construction shows that forbidding triangles is insufficient to guarantee bounded chromatic number for a wide class of planar objects, suggesting that stronger geometric restrictions (e.g., fatness, bounded aspect ratio, or limited orientation) are necessary for positive results. They also propose extending the method to higher dimensions, where the relationship between ω and χ for intersection graphs of boxes, ellipsoids, or other convex bodies remains largely unexplored. Finally, the paper highlights the potential of using online‑algorithmic adversarial techniques as a systematic tool for generating extremal examples in geometric graph theory.

Overall, the work settles an open problem, provides a versatile construction applicable to many familiar shapes, and bridges two previously separate strands of combinatorial geometry and online algorithm analysis.