Coloring curves that cross a fixed curve

We prove that for every integer $t\geq 1$, the class of intersection graphs of curves in the plane each of which crosses a fixed curve in at least one and at most $t$ points is $\chi$-bounded. This is essentially the strongest $\chi$-boundedness result one can get for this kind of graph classes. As a corollary, we prove that for any fixed integers $k\geq 2$ and $t\geq 1$, every $k$-quasi-planar topological graph on $n$ vertices with any two edges crossing at most $t$ times has $O(n\log n)$ edges.


Overview.
A curve is a homeomorphic image of the real interval [0, 1] in the plane. The intersection graph of a family of curves has these curves as vertices and the intersecting pairs of curves as edges. Combinatorial and algorithmic aspects of intersection graphs of curves, known as string graphs, have been attracting researchers for decades. A significant part of this research has been devoted to understanding classes of string graphs that are χ-bounded, which means that every graph G in the class satisfies χ(G) f (ω(G)) for some function f : N → N. Here, χ(G) and ω(G) denote the chromatic number and the clique number (the maximum size of a clique) of G, respectively. Recently, Pawlik et al. [24,25] proved that the class of all string graphs is not χ-bounded. However, all known constructions of string graphs with small clique number and large chromatic number require a lot of freedom in placing curves around in the plane.
What restrictions on placement of curves lead to χ-bounded classes of intersection graphs? McGuinness [19,20] proposed studying families of curves that cross a fixed curve exactly once. This initiated a series of results culminating in the proof that the class of intersection graphs of such families is indeed χ-bounded [26]. By contrast, the class of intersection graphs of curves each crossing a fixed curve at least once is equal to the class of all string graphs and therefore is not χ-bounded. We prove an essentially farthest possible generalization of the former result, allowing curves to cross the fixed curve at least once and at most t times, for any bound t.

Theorem 1. For every integer t 1, the class of intersection graphs of curves each crossing a fixed curve in at least one and at most t points is χ-bounded.
Additional motivation for Theorem 1 comes from its application to bounding the number of edges in so-called k-quasi-planar graphs, which we discuss at the end of this introduction.
Context. Colorings of intersection graphs of geometric objects have been investigated since the 1960s, when Asplund and Grünbaum [3] proved that intersection graphs of axis-parallel rectangles in the plane satisfy χ = O(ω 2 ) and conjectured that the class of intersection graphs of axis-parallel boxes in R d is χ-bounded for every integer d 1. A few years later Burling [5] discovered a surprising construction of triangle-free intersection graphs of axis-parallel boxes in R 3 with arbitrarily large chromatic number. Since then, the upper bound of O(ω 2 ) and the trivial lower bound of Ω(ω) on the maximum possible chromatic number of a rectangle intersection graph have been improved only in terms of multiplicative constants [11,13].
Another classical example of a χ-bounded class of geometric intersection graphs is provided by circle graphs-intersection graphs of chords of a fixed circle. Gyárfás [10] proved that circle graphs satisfy χ = O(ω 2 4 ω ). The best known upper and lower bounds on the maximum possible chromatic number of a circle graph are O(2 ω ) [14] and Ω(ω log ω) [12,13].
McGuinness [19,20] proposed investigating the problem in a setting that allows much more general geometric shapes but restricts the way how they are arranged in the plane. In [19], he proved that the class of intersection graphs of L-shapes crossing a fixed horizontal line is χ-bounded. Families of L-shapes in the plane are simple, which means that any two members of the family intersect in at most one point. McGuinness [20] also showed that triangle-free intersection graphs of simple families of curves each crossing a fixed line in exactly one point have bounded chromatic number. Further progress in this direction was made by Suk [27], who proved that simple families of x-monotone curves crossing a fixed vertical line give rise to a χbounded class of intersection graphs, and by Lasoń et al. [17], who reached the same conclusion without assuming that the curves are x-monotone. Finally, in [26], we proved that the class of intersection graphs of curves each crossing a fixed line in exactly one point is χ-bounded. These results remain valid if the fixed straight line is replaced by a fixed curve [28].
The class of string graphs is not χ-bounded. Pawlik et al. [24,25] showed that Burling's construction for boxes in R 3 can be adapted to provide a construction of triangle-free intersection graphs of straight-line segments (or geometric shapes of various other kinds) with chromatic number growing as fast as Θ(log log n) with the number of vertices n. It was further generalized to a construction of string graphs with clique number ω and chromatic number Θ ω ((log log n) ω−1 ) [16]. The best known upper bound on the chromatic number of string graphs in terms of the number of vertices is (log n) O(log ω) , proved by Fox and Pach [8] using a separator theorem for string graphs due to Matoušek [18]. For intersection graphs of segments and, more generally, x-monotone curves, upper bounds of the form χ = O ω (log n) follow from the above-mentioned results in [27] and [26] via recursive halving. Upper bounds of the form χ = O ω ((log log n) f (ω) ) (for some function f : N → N) are known for very special classes of string graphs: rectangle overlap graphs [15,16] and subtree overlap graphs [16]. The former still allow the triangle-free construction with χ = Θ(log log n) and the latter the construction with χ = Θ ω ((log log n) ω−1 ).

Quasi-planarity.
A topological graph is a graph with a fixed curvilinear drawing in the plane. For k 2, a k-quasi-planar graph is a topological graph with no k pairwise crossing edges. In particular, a 2-quasi-planar graph is just a planar graph. It is conjectured that k-quasi-planar graphs with n vertices have O k (n) edges [4,23]. For k = 2, this asserts a well-known property of planar graphs. The conjecture is also verified for k = 3 [2,22] and k = 4 [1], but it remains open for k 5. The best known upper bounds on the number of edges in a k-quasi-planar graph are n(log n) O(log k) in general [7,8], O k (n log n) for the case of x-monotone edges [29], O k (n log n) for the case that any two edges intersect at most once [28], and 2 α(n) ν n log n for the case that any two edges intersect in at most t points, where α is the inverse Ackermann function and ν depends on k and t [28]. We apply Theorem 1 to improve the last bound to O k,t (n log n).

Theorem 2.
Every k-quasi-planar topological graph G on n vertices such that any two edges of G intersect in at most t points has at most µ k,t n log n edges, where µ k,t depends only on k and t.
The proof follows the same lines as the proof in [28] for the case t = 1 (see Section 3).

Proof of Theorem 1
Setup. We let N denote the set of positive integers. Graph-theoretic terms applied to a family of curves F have the same meaning as applied to the intersection graph of F. In particular, the chromatic number of F, denoted by χ(F), is the minimum number of colors in a proper coloring of F (a coloring that distinguishes pairs of intersecting curves), and the clique number of F, denoted by ω(F), is the maximum size of a clique in F (a set of pairwise intersecting curves in F).
Theorem 1 (rephrased). For every t ∈ N, there is a non-decreasing function f t : N → N with the following property: for any fixed curve c 0 , every family F of curves each intersecting c 0 in at least one and at most t points satisfies χ(F) f t (ω(F)).
We do not state any explicit bound on the function f t above, because it highly depends on the bound on the function f in Theorem 13 (one of our main tools), and no explicit bound on that function is provided in [6]. We assume (implicitly) that the intersection points of all curves c ∈ F with c 0 considered in Theorem 1 are distinct and each of them is a proper crossing, which means that c passes from one to the other side of c 0 in a sufficiently small neighborhood of the intersection point. This assumption is without loss of generality, as it can be guaranteed by appropriate small perturbations of the curves that do not influence the intersection graph.
Initial reduction. We start by reducing Theorem 1 to a somewhat simpler and more convenient setting. We fix a horizontal line in the plane and call it the baseline. The upper closed half-plane determined by the baseline is denoted by H + . A 1-curve is a curve in H + that has one endpoint (called the basepoint of the 1-curve) on the baseline and does not intersect the baseline in any other point. Intersection graphs of 1-curves are known as outerstring graphs and form a χ-bounded class of graphs-this result, due to the authors, is the starting point of the proof of Theorem 1.
An even-curve is a curve that has both endpoints above the baseline and a positive even number of intersection points with the baseline, each of which is a proper crossing. For t ∈ N, a 2t-curve is an even-curve that intersects the baseline in exactly 2t points. A basepoint of an even-curve c is an intersection point of c with the baseline. Like above, we assume (implicitly, without loss of generality) that the basepoints of all even-curves in any family that we consider are distinct. Every even-curve c determines two 1-curves-the two parts of c from an endpoint to the closest basepoint along c. They are called the 1-curves of c and denoted by L(c) and R(c) so that the basepoint of L(c) lies to the left of the basepoint of R(c) on the baseline (see Figure 1). A family F of even-curves is an LR-family if every intersection between two curves c 1 , c 2 ∈ F is an intersection between L(c 1 ) and R(c 2 ) or between L(c 2 ) and R(c 1 ). The main effort in this paper goes to proving the following statement on LR-families of even-curves.

Theorem 4.
There is a non-decreasing function f : N → N such that every LR-family F of even-curves satisfies χ(F) f (ω(F)).
Theorem 4 makes no assumption on the maximum number of intersection points of an evencurve with the baseline. We derive Theorem 1 from Theorem 4 in two steps, first proving the following lemma, and then showing that Theorem 1 is essentially a special case of it.
We establish the base case for t = 1 and the induction step for t 2 simultaneously. Namely, fix an integer t 1, and let F be as in the statement of the lemma. For every 2t-curve c ∈ F, enumerate the endpoints and basepoints of c as p 0 (c), . . . , p 2t+1 (c) in their order along c so that p 0 (c) and p 1 (c) are the endpoints of L(c) while p 2t (c) and p 2t+1 (c) are the endpoints of R(c). Build two families of curves F 1 and F 2 putting the part of c from p 0 (c) to p 2t−1 (c) to F 1 and the part of c from p 2 (c) to p 2t+1 (c) to F 2 for every c ∈ F. If t = 1, then F 1 and F 2 are families of 1-curves. If t 2, then F 1 and F 2 are equivalent to families of 2(t − 1)curves, because the curve in F 1 or F 2 obtained from a 2t-curve c ∈ F can be shortened a little at p 2t−1 (c) or p 2 (c), respectively, losing that basepoint but no intersection points with other curves. Therefore, by Theorem 3 or the induction hypothesis, we have For c ∈ F and k ∈ {1, 2}, let φ k (c) be the color of the curve obtained from c in an optimal proper coloring of F k . Every subfamily of F on which φ 1 and φ 2 are constant is an LR-family and therefore, by Theorem 4 and monotonicity of f , has chromatic number at most f (ω(F)). We conclude that χ(F) χ(F 1 )χ(F 2 )f (ω(F)) f 2 t−1 (ω(F))f (ω(F)) = f t (ω(F)). A closed curve is a homeomorphic image of a unit circle in the plane. For a closed curve γ, the Jordan curve theorem asserts that the set R 2 γ consists of two arc-connected components, one of which is bounded and denoted by int γ and the other is unbounded and denoted by ext γ.
Proof of Theorem 1 from Theorem 4. We elect to present this proof in an intuitive rather than rigorous way. Let F be a family of curves each intersecting c 0 in at least one and at most t points. Let γ 0 be a closed curve surrounding c 0 very closely so that γ 0 intersects every curve in F in exactly 2t points (winding if necessary to increase the number of intersections) and all endpoints of curves in F and intersection points of pairs of curves in F lie in ext γ 0 . We apply geometric inversion to obtain an equivalent family of curves F and a closed curve γ 0 with the same properties except that all endpoints of curves in F and intersection points of pairs of curves in F lie in int γ 0 . It follows that some part of γ 0 lies in the unbounded component of R 2 F . We "cut" γ 0 there and "unfold" it into the baseline, transforming F into an equivalent family F of 2t-curves all endpoints of which and intersection points of pairs of which lie above the baseline. The "equivalence" of F, F , and F means in particular that the intersection graphs of F, F , and F are isomorphic, so the theorem follows from Lemma 5 (and thus Theorem 4).
A statement analogous to Theorem 4 fails for families of objects each consisting of two 1curves only, without the "middle part" connecting them. Specifically, we define a double-curve as a set X ⊂ H + that is a union of two disjoint 1-curves, denoted by L(X) and R(X) so that the basepoint of L(X) lies to the left of the basepoint of R(X), and we call a family X of double-curves an LR-family if every intersection between two double-curves X 1 , X 2 ∈ X is an intersection between L(X 1 ) and R(X 2 ) or between L(X 2 ) and R(X 1 ).

Theorem 6. There exist triangle-free LR-families of double-curves with arbitrarily large chromatic number.
The proof of Theorem 6 is an easy adaptation of the construction from [24,25] and is presented in detail in Section 4. The rest of this section is devoted to the proof of Theorem 4.
Overview of the proof of Theorem 4. Recall the assertion of Theorem 4: the LR-families of even-curves are χ-bounded. The proof is quite long and technical, so we find it useful to provide a high-level overview of its structure. The proof will be presented via a series of reductions. First, we will reduce Theorem 4 to the following statement (Lemma 7): the LR-families of 2-curves are χ-bounded. This statement will be proved by induction on the clique number. Specifically, we will prove the following as the induction step: if every LR-family ζ is a constant depending only on k and ξ. The only purpose of the induction hypothesis is to guarantee that if ω(F) k and c ∈ F, then the family of 2-curves in F {c} that intersect c has chromatic number at most ξ. For notational convenience, LR-families of 2-curves with the latter property will be called ξ-families. We will thus reduce the problem to the following statement (Lemma 9): the ξ-families are χ-bounded, where the χ-bounding function depends on ξ.
We will deal with ξ-families via a series of technical lemmas of the following general form: every ξ-family with sufficiently large chromatic number contains a specific configuration of curves. Two kinds of such configurations are particularly important: (a) a large clique, and (b) a 2-curve c and a subfamily F with large chromatic number such that all basepoints of 2-curves in F lie between the basepoints of c. At the core of the argument are the proofs that • every ξ-family with sufficiently large chromatic number contains (a) or (b) (Lemma 16), • assuming the above, every ξ-family with sufficiently large chromatic number contains (a). Combined, they complete the argument. Since the two proofs are almost identical, we introduce one more reduction-to (ξ, h)-families (Lemma 15). A (ξ, h)-family is just a ξ-family that satisfies an additional technical condition that allows us to deliver both proofs at once.
More notation and terminology. Let ≺ denote the left-to-right order of points on the baseline (p 1 ≺ p 2 means that p 1 is to the left of p 2 ). For convenience, we also use the notation ≺ for curves intersecting the baseline (c 1 ≺ c 2 means that every basepoint of c 1 is to the left of every basepoint of c 2 ) and for families of such curves (C 1 ≺ C 2 means that c 1 ≺ c 2 for any c 1 ∈ C 1 and c 2 ∈ C 2 ). For a family C of curves intersecting the baseline (even-curves or 1-curves) and two 1-curves x and y, let A cap-curve is a curve in H + that has both endpoints on the baseline and does not intersect the baseline in any other point. It follows from the Jordan curve theorem that for every capcurve γ, the set H + γ consists of two arc-connected components, one of which is bounded and denoted by int γ and the other is unbounded and denoted by ext γ.
Reduction to LR-families of 2-curves. We will reduce Theorem 4 to the following statement on LR-families of 2-curves, which is essentially a special case of Theorem 4.

Lemma 7. There is a non-decreasing function
A component of a family of 1-curves S is an arc-connected component of S (the union of all curves in S). The following easy but powerful observation reuses an idea from [17,20,27].
Proof. Let G be an auxiliary graph where the vertices are the components of L(F) ∪ R(F) and the edges are the pairs Since F is an LR-family, the curves in F can intersect only within the components of L(F) ∪ R(F). It follows that G is planar and thus 4-colorable. Fix a proper 4-coloring of G, and assign the color of a component V to every curve c ∈ F with Let c 1 , c 2 ∈ F 1 . We claim that the intervals I(c 1 ) and I(c 2 ) are nested or disjoint. Suppose they are neither nested nor disjoint. The components of L(F) ∪ R(F) are disjoint from the curves of the form M (c) with c ∈ F except at common basepoints. For k ∈ {1, 2}, since L(c k ) and R(c k ) belong to the same component of L(F) ∪ R(F), the basepoints of L(c k ) and R(c k ) can be connected by a cap-curve γ k disjoint from M (c) for every c ∈ F except at the endpoints of M (c) when c = c k . We assume (without loss of generality) that γ 1 and γ 2 intersect in a finite number of points and each of their intersection points is a proper crossing. Since the intervals I(c 1 ) and I(c 2 ) are neither nested nor disjoint, the basepoints of L(c 2 ) and R(c 2 ) lie one in int γ 1 and the other in ext γ 1 . This implies that γ 1 and γ 2 intersect in an odd number of points, by the Jordan curve theorem. For k ∈ {1, 2}, letγ k be the closed curve obtained as the union of γ k and M (c k ). It follows thatγ 1 andγ 2 intersect in an odd number of points and each of their intersection points is a proper crossing, which is a contradiction to the Jordan curve theorem.
Transform F 1 into a family of 2-curves F 1 replacing the part M (c) of every 2-curve c ∈ F 1 by the lower semicircle connecting the endpoints of M (c). Since the intervals I(c) with c ∈ F 1 are pairwise nested or disjoint, these semicircles are pairwise disjoint. Consequently, F 1 is an LR-family. Since the intersection graphs of F 1 and F 1 are isomorphic, Lemma 7 implies Reduction to ξ-families. For ξ ∈ N, a ξ-family is an LR-family of 2-curves F with the following property: for every 2-curve c ∈ F, the family of 2-curves in F {c} that intersect c has chromatic number at most ξ. We reduce Lemma 7 to the following statement on ξ-families.

Lemma 9.
For any ξ, k ∈ N, there is a constant ζ ∈ N such that every ξ-family F with ω(F) k satisfies χ(F) ζ.
Proof of Lemma 7 from Lemma 9. Let f (1) = 1. For k 2, let f (k) be the constant claimed by Lemma 9 such that every f (k − 1)-family F with ω(F) k satisfies χ(F) f (k). Let k = ω(F), and proceed by induction on k to prove χ(F) f (k). Clearly, if k = 1, then χ(F) = 1. For the induction step, assume k 2. For every c ∈ F, the family of 2-curves in F {c} that intersect c has clique number at most k−1 and therefore, by the induction hypothesis, has chromatic number at most f (k − 1). That is, F is an f (k − 1)-family, and the definition of f yields χ(F) f (k).
Dealing with ξ-families. First, we establish the following special case of Lemma 9.
The proof of Lemma 10 is essentially the same as the proof of Lemma 19 in [28]. We need the following elementary lemma, which was also used in various forms in [17,19,20,26,27]. We include its proof, as we will later extend it when proving Lemma 12.  We conclude that χ {c i+1 , . . . , c j−1 } 2ξ + 1, which is a contradiction.
Finally, suppose χ n−2 i=0 n j=i+2 F i,j > 6α. It follows that χ i∈I n j=i+2 F i,j > 3α, where I = {i ∈ {0, . . . , n − 2} : i ≡ 0 (mod 2)} or I = {i ∈ {0, . . . , n − 2} : i ≡ 1 (mod 2)}. Consider an auxiliary graph G with vertex set I and edge set {ij : i, j ∈ I, i < j, and F i,j−1 ∪ F i,j = ∅}. If there were two edges i 1 j 1 and i 2 j 2 in G with i 1 < i 2 < j 1 < j 2 , then their witnessing 2-curves, one from F i 1 ,j 1 −1 ∪ F i 1 ,j 1 and the other from F i 2 ,j 2 −1 ∪ F i 2 ,j 2 , would intersect below the baseline, which is impossible. This shows that G is an outerplanar graph, and thus χ(G) 3. Fix a proper 3-coloring of G, and use the color of i on every 2-curve in n j=i+2 F i,j for every i ∈ I, partitioning the family i∈I n j=i+2 F i,j into 3 color classes. At least one such color class H satisfies χ(H) > α. To conclude, for any two intersecting 1-curves x ∈ R(H) and y ∈ L(H), we have x ∈ R(F i,j ) and y ∈ L(F r,s ) for some indices i, r ∈ I, j ∈ {i + 2, . . . , n}, and s ∈ {r + 2, . . . , n} such that j / ∈ {r − 1, r} (otherwise ir would be an edge of G), j = r + 1 (otherwise two 2-curves, one from F i,r+1 and one from F r,s , would intersect below the baseline), and thus |j − r| 2. Lemma 2 in [26] asserts that for every family of 1-curves S with at least one intersecting pair, there are a cap-curve γ and a subfamily T ⊆ S with χ(T ) χ(S)/2 such that every 1-curve in T is entirely contained in int γ and intersects some 1-curve in S that intersects γ (equivalently, ext γ). The proof follows a standard idea, originally due to Gyárfás [10], to choose T as one of the sets of 1-curves at a fixed distance from an appropriately chosen 1-curve in the intersection graph of S. However, this method fails to imply an analogous statement for 2-curves. We will need a more powerful tool-part of the recent series of works on induced subgraphs that must be present in graphs with sufficiently large chromatic number. Proof. Let f (α) = f 1 (3α + 5ξ + 5), where f 1 is the function claimed by Theorem 13. Let F be a ξfamily with χ(F) > f (α). It follows that there is a 2-curve c ∈ F such that the family of curves within distance at most 2 from c in the intersection graph of F has chromatic number greater than 3α + 5ξ + 5. For k ∈ {1, 2}, let F k be the 2-curves in F at distance exactly k from c in the intersection graph of F. We have χ({c }∪F 1 ∪F 2 ) > 3α +5ξ +5 (by Theorem 13) and χ(F 1 ) ξ (because F is a ξ-family), so χ(F 2 ) > 3α + 4ξ + 4. We have Since χ(F 2 ) > 3α + 4ξ + 4 and χ(G 4 ) 4ξ + 4 (by Lemma 10), we have χ(G k ) > α for some k ∈ {1, 2, 3}. Since neither basepoint of c lies on the segment I(G k ), there is a cap-curve γ with L(c ), R(c ) ⊂ ext γ and L(c), R(c) ⊂ int γ for all c ∈ G k . The lemma follows with G = G k . For ξ ∈ N and a function h : N → N, a (ξ, h)-family is a ξ-family F with the following additional property: for every α ∈ N and every subfamily G ⊆ F with χ(G) > h(α), there is a subfamily H ⊆ G with χ(H) > α such that every 2-curve in F with a basepoint on I(H) has both basepoints on I(G). We will prove the following lemma.

Reduction to (ξ, h)-families.
. Let F be a ξ-family with ω(F) k and χ(F(I(c))) α for every c ∈ F. We show that F is a (ξ, h α )-family, which then implies χ(F) f (α). To this end, consider a subfamily We show that F is a (ξ, h)-family, which then implies χ(F) ζ. To this end, consider a subfamily G ⊆ F with χ(G) > h(α) for some α ∈ N. Lemma 16 yields a 2-curve c ∈ G such that χ(G(I(c))) > α. Every 2-curve in F with a basepoint on I(c) has both basepoints on I(c), otherwise it would intersect c below the baseline. Therefore, the condition on F being a (ξ, h)-family is satisfied with H = G(I(c)).
Dealing with (ξ, h)-families. The rest of this section is devoted to the proof of Lemma 15. Its structure and principal ideas are based on those of the proof of Theorem 3 presented in [26]. For each forthcoming lemma, we provide a reference to its counterpart in [26].
A skeleton is a pair (γ, U) such that γ is a cap-curve and U is a family of pairwise disjoint 1-curves each of which has one endpoint (other than the basepoint) on γ and all the remaining part in int γ (see Figure 3). For a family of 1-curves S, a skeleton (γ, U) is an S-skeleton if every 1-curve in U is a subcurve of some 1-curve in S. A family of 2-curves G is supported by a skeleton (γ, U) if every 2-curve c ∈ G satisfies L(c), R(c) ⊂ int γ and intersects some 1-curve in There is a cap-curve ν ⊆ V connecting the two endpoints of the segment I(G V ). Suppose there is a 2-curve c ∈ F ext with both basepoints on I(G V ). If L(c) intersects ext γ, then the part of L(c) from the basepoint to the first intersection point with γ, which is a 1-curve in U L , intersects ν (as ν ⊆ V ⊂ int γ) and thus a curve in G (as V is a component of G ); this implies G ∩ G L = ∅, which is a contradiction. An analogous contradiction is reached if R(c) intersects ext γ. This shows that no curve in F ext has both basepoints on I(G V ).
Since Proof 1 (α, 4ξ))), where f 1 is the function claimed by Lemma 17. Let F be a (ξ, h)-family with χ(F) > f (α). Suppose for the sake of contradiction that every subfamily of F supported by an L(F)-skeleton or an R(F)-skeleton has chromatic number at most α. Let F 0 = F. Apply Lemma 17 (and the second conclusion thereof) three times to find families F 1 , F 2 , F 3 , S 1 , S 2 , and S 3 with the following properties: There are indices i and j with 1 i < j 3 such that S i and S j are of the same "type": either . Assume for the rest of the proof that S i ⊆ R(F i−1 ) and S j ⊆ R(F j−1 ); the argument for the other case is analogous. Let S L = {s ∈ S j : s ≺ F j }, S R = {s ∈ S j : F j ≺ s}, F L be the 2-curves in F j that intersect some 1-curve in S L , and F R be those that intersect some 1-curve in S R . Thus F L ∪ F R = F j . This and χ(F j ) χ(F 3 ) > 4ξ yield χ(F L ) > 2ξ or χ(F R ) > 2ξ. Assume for the rest of the proof that χ(F L ) > 2ξ; the argument for the other case is analogous.
Let S min L be an inclusion-minimal subfamily of S L subject to the condition that L(c) intersects some 1-curve in S min L for every 2-curve c ∈ F L . Let s be the 1-curve in S min L with rightmost basepoint, and let F L = {c ∈ F L : L(c) intersects s }. Since F is a ξ-family, we have χ(F L ) ξ. By minimality of S min L , the family F L contains a 2-curve c disjoint from every 1-curve in S min L other than s . Since c ∈ F j ⊆ F i and F i is supported from outside by S i , there is a 1-curve s i ∈ S i that intersects L(c ). We show that every 2-curve in F L F L intersects s i . Let c ∈ F L F L , and let s be a 1-curve in S min L that intersects L(c). We have s = s , as c / ∈ F L . There is a cap-curve ν ⊆ s ∪ L(c). Since s ≺ s ≺ L(c) and s intersects neither s nor L(c), we have s ⊂ int ν. Since L(c ) intersects s but neither s nor L(c), we also have L(c ) ⊂ int ν. Since s ∈ S j ⊆ R(F i ) and s i ≺ F i or F i ≺ s i , the basepoint of s i lies in ext ν. Since s i intersects L(c ) and L(c ) ⊂ int ν, the 1-curve s i intersects ν and thus L(c). This shows that every 2-curve in F L F L intersects s i . This and the assumption that F is a ξ-family yield χ(F L F L ) ξ. We conclude that χ(F L ) χ(F L ) + χ(F L F L ) 2ξ, which is a contradiction.
A chain of length n is a sequence (a 1 , b 1 ), . . . , (a n , b n ) of pairs of 2-curves such that Let ). Let F be a (ξ, h)-family with χ(F) > f (n + 1). We claim that F contains a chain of length n + 1.
Let F 0 = F. Lemma 18 applied three times provides families of 2-curves F 1 , F 2 , F 3 and skeletons (γ 1 , U 1 ), (γ 2 , U 2 ), (γ 3 , U 3 ) with the following properties: There are indices i and j with 1 i < j 3 such that the skeletons (γ i , U i ) and (γ j , U j ) are of the same "type": either an L(F i−1 )-skeleton and an L(F j−1 )-skeleton or an R(F i−1 )-skeleton and an R(F j−1 )-skeleton. Assume for the rest of the proof that (γ i , U i ) is an L(F i−1 )-skeleton and (γ j , U j ) is an L(F j−1 )-skeleton; the argument for the other case is analogous. By Lemma 12, since χ(F j ) χ(F 3 ) > β, there is a subfamily H ⊆ F j such that χ(H) > f (n) and χ(F j (x, y)) > h(2ξ) + 4ξ + 2 for any two intersecting 1-curves x, y ∈ L(H) ∪ R(H). Since there is a chain (a 1 , b 1 ), . . . , (a n , b n ) of length n in H. Let x and y be the 1-curves R(a n ) and L(b n ) ordered so that x ≺ y. Since they intersect, we have χ(F j (x, y)) > h(2ξ)+4ξ +2.
Since , y), then c intersects the 1-curve in U i with rightmost basepoint to the left of the basepoint of x (if such a 1-curve exists) or the 1-curve in U i with leftmost basepoint to the right of the basepoint of y (if such a 1-curve exists). This and the fact that F is a ξ-family imply χ(F j (x, y) G) 2ξ and thus χ(G) χ(F j (x, y)) − 2ξ > h(2ξ) + 2ξ + 2.
The rest of the argument is illustrated in Figure 4. Let u L and u R be the curves in U i (x, y) with leftmost and rightmost basepoints, respectively. Every 1-curve in U i (x, y) lies in the closed region K bounded by u L , u R , the segment of the baseline between the basepoints of u L and u R , and the part of γ i between its intersection points with u L and u R . Since F is a ξ-family, the 2-curves in G intersecting u L or u R have chromatic number at most 2ξ. Every other 2curve c ∈ G satisfies L(c) ⊂ K or R(c) ⊂ K. Those for which L(c) ⊂ K but R(c) ⊂ K satisfy R(c) ∩ K = ∅ and therefore are disjoint from each other. Similarly, those for which R(c) ⊂ K but L(c) ⊂ K are disjoint from each other. Let G = {c ∈ G : L(c) ⊂ K and R(c) ⊂ K}. It follows that χ(G G ) 2ξ + 2 and thus χ( Since F is a (ξ, h)-family, there is a subfamily H ⊆ G with χ(H ) > 2ξ such that every 2curve c ∈ F with a basepoint on I(H ) satisfies u L ≺ c ≺ u R . Since H ⊆ F j and F j is supported by (γ j , U j ), every 2-curve in H intersects some 1-curve in U j . If a 2-curve c ∈ H intersects no 1-curve in U j (I(H )), then c intersects the 1-curve in U j with rightmost basepoint to the left of I(H ) (if such a 1-curve exists) or the 1-curve in U j with leftmost basepoint to the right of I(H ) (if such a 1-curve exists). Since F is a ξ-family, the 2-curves in H intersecting at least one of these two 1-curves have chromatic number at most 2ξ. Therefore, since χ(H ) > 2ξ, some 2-curve in H intersects a 1-curve in U j (I(H )). In particular, the family U j (I(H )) is non-empty.
Let u ∈ U j (I(H )). The 1-curve u is a subcurve of L(c ) for some 2-curve c ∈ F j−1 . The fact that the basepoint of L(c ) lies on I(H ) and the property of H imply u L ≺ c ≺ u R . Since c ∈ F j−1 ⊆ F i and F i is supported by (γ i , U i ), the 1-curve R(c ) intersects a 1-curve u ∈ U i , which can be chosen so that u L u u R , because c ⊂ int γ i and both basepoints of c lie in K. Let a n+1 = c and b n+1 be the 2-curve in F i−1 such that u is a subcurve of L(b n+1 ). Thus x ≺ {R(a n+1 ), L(b n+1 )} ≺ y. For 1 t n, the facts that the 1-curves R(a t ) and L(b t ) intersect, they are both contained in int γ j (as a t , b t ∈ F j ), the basepoint of u lies between the basepoints of R(a t ) and L(b t ), and u intersects γ j imply that u and therefore L(a n+1 ) intersects R(a t ). We conclude that (a 1 , b 1 ), . . . , (a n+1 , b n+1 ) is a chain of length n + 1.

Proof of Theorem 2
Lemma 20 (Fox,Pach,Suk [9,Lemma 3.2]). For every t ∈ N, there is a constant ν t > 0 such that every family of curves F any two of which intersect in at most t points has subfamilies F 1 , . . . , F d ⊆ F (where d is arbitrary) with the following properties: • for 1 i d, there is a curve c i ∈ F i intersecting all curves in F i {c i }, • for 1 i < j d, every curve in F i is disjoint from every curve in F j , • |F 1 ∪ · · · ∪ F d | ν t |F|/ log |F|.
Proof of Theorem 2. Let F be a family of curves obtained from the edges of G by shortening them slightly so that they do not intersect at the endpoints but all other intersection points are preserved. If follows that ω(F) k − 1 (as G is k-quasi-planar) and any two curves in F intersect in at most t points. Let ν t , F 1 , . . . , F d , and c 1 , . . . , c d be as claimed by Lemma 20. For 1 i d, since ω(F i {c i }) ω(F)−1 k −2, Theorem 1 yields χ(F i {c i }) f t (k −2). Thus χ(F 1 ∪ · · · ∪ F d ) f t (k − 2) + 1. For every color class C in a proper coloring of F 1 ∪ · · · ∪ F d with f t (k − 2) + 1 colors, the vertices of G and the curves in C form a planar topological graph, and thus |C| < 3n. Thus |F 1 ∪· · ·∪F d | < 3(f t (k −2)+1)n. This, the third property in Lemma 20, and the fact that |F| < n 2 yield |F| < 3ν −1 t (f t (k − 2) + 1)n log |F| < 6ν −1 t (f t (k − 2) + 1)n log n.

Proof of Theorem 6
Proof of Theorem 6. A probe is a section of H + bounded by two vertical rays starting at the baseline. We use induction to construct, for every positive integer k, an LR-family X k of doublecurves and a family P k of pairwise disjoint probes with the following properties: (1) every probe in P k is disjoint from L(X) for every double-curve X ∈ X k , (2) for every probe P ∈ P k , the double-curves in X k intersecting P are pairwise disjoint, (3) X k is triangle-free, that is, ω(X k ) 2, (4) for every proper coloring of X k , there is a probe P ∈ P k such that at least k distinct colors are used on the double-curves in X k intersecting P . This is enough for the proof of theorem, because the last property implies χ(X k ) k. For a pair (X k , P k ) satisfying the conditions above and a probe P ∈ P k , let X k (P ) denote the set of double-curves in X k intersecting P . For the base case k = 1, we let X 1 = {X} and P 1 = {P }, where X and P look as follows:

P L(X) R(X)
It is clear that the conditions (1)-(4) are satisfied. For the induction step, we assume k 1 and construct the pair (X k+1 , P k+1 ) from (X k , P k ). Let (X , P) be a copy of (X k , P k ). For every probe P ∈ P, put another copy (X P , P P ) of (X k , P k ) inside P below the intersections of P with the double-curves in X (P ). Then, for every probe P ∈ P and every probe Q ∈ P P , let a double-curve X P Q and probes A P Q and B P Q look as follows: In particular, X P Q intersects the double-curves in X P (Q), A P Q intersects the double-curves in X (P ) ∪ X P (Q), and B P Q intersects the double-curves in X (P ) ∪ {X P Q }. Let The conditions (1) and (2) clearly hold for (X k+1 , P k+1 ), and (2) for (X k , P k ) implies (3) for (X k+1 , P k+1 ). To see that (4) holds for (X k+1 , P k+1 ) and k + 1, consider a proper coloring φ of X k+1 . Let φ(X) denote the color of a double-curve X ∈ X k+1 and φ(Y) denote the set of colors used on a subset Y ⊆ X k+1 . By (4) applied to (X , P), there is a probe P ∈ P such that |φ(X (P ))| k. By (4) applied to (X P , P P ), there is a probe Q ∈ P P such that |φ(X P (Q))| k. Since X P Q intersects the double-curves in X P (Q), we have φ(X P Q ) / ∈ φ(X P (Q)). If φ(X (P )) = φ(X P (Q)), then X k+1 (A P Q ) = X (P ) ∪ X P (Q) yields |φ(X k+1 (A P Q ))| = |φ(X (P )) ∪ φ(X P (Q))| k + 1. If φ(X (P )) = φ(X P (Q)), then X k+1 (B P Q ) = X (P ) ∪ {X P Q } and φ(X P Q ) / ∈ φ(X (P )) yield |φ(X k+1 (B P Q ))| = |φ(X (P ))+1| k +1. This shows that (4) holds for (X k+1 , P k+1 ) and k +1.