Further Consequences of the Colorful Helly Hypothesis

Let F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document} be a family of convex sets in Rd,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^d,$$\end{document} which are colored with d+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d+1$$\end{document} colors. We say that F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document} satisfies the Colorful Helly Property if every rainbow selection of d+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d+1$$\end{document} sets, one set from each color class, has a non-empty common intersection. The Colorful Helly Theorem of Lovász states that for any such colorful family F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document} there is a color class Fi⊂F,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}_i\subset \mathcal {F},$$\end{document} for 1≤i≤d+1,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$1\le i\le d+1,$$\end{document} whose sets have a non-empty intersection. We establish further consequences of the Colorful Helly hypothesis. In particular, we show that for each dimension d≥2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d\ge 2$$\end{document} there exist numbers f(d) and g(d) with the following property: either one can find an additional color class whose sets can be pierced by f(d) points, or all the sets in F\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {F}$$\end{document} can be crossed by g(d) lines.


Helly-Type Theorems
Let F be a finite family of convex sets in R d . We say that a collection X of geometric objects (e.g., points, lines, or k-flats-k-dimensional affine subspaces of R d ) is a transversal to F, or that F can be pierced or crossed by X , if each set of F is intersected by some member of X . For an integer j we use the symbol F j to denote the collection of subfamilies of F of size j.
The 1913 theorem of Helly [16] states that a finite family F of convex sets has a non-empty intersection (i.e., F can be pierced by a single point) if and only if each of its subsets F ⊂ F of size at most d + 1 can be pierced by a point.
• Given that a significant fraction of the (d +1)-tuples F ∈ F d+1 have a non-empty intersection, can F, or at least some fixed fraction of its members, be pierced by constantly many points?
The first question has been settled to the negative already for k = 1. For instance, Santaló [21] and Danzer [11] observed that for any n ≥ 3 there are families F of n convex sets in R 2 such that any n − 1 of the sets can be crossed by a single line transversal while no such transversal exists for F. Nevertheless, Alon and Kalai [2] show that the following almost-Helly property holds for k = d − 1: if every d + 1 (or fewer) of the sets of F can be crossed by a hyperplane, then F admits a transversal by h hyperplanes, where the number h = h(d) depends only on the dimension d.
While the properties of hyperplane transversals largely resemble those of point transversals, this is not the case for transversals by k-flats of intermediate dimensions 1 ≤ k ≤ d − 2. For example, Alon et al. [3] showed that for every integers d ≥ 3, m and n 0 ≥ m + 4 there is a family of at least n 0 convex sets such that any m of the sets can be crossed by a line but no m + 4 of them can; this phenomenon can be largely attributed to the complex topological structure of the space of transversal k-flats.
The second question gave rise to a plethora of inter-related results in discrete geometry and topological combinatorics. Theorem 1.1 (Fractional Helly's Theorem) For any d ≥ 1 and α > 0 there is a number β = β(α, d) > 0 with the following property: for every finite family F of convex sets in R d such that at least α |F | d+1 of the (d + 1)-subsets F ∈ F d+1 have non-empty intersection, there is a point which pierces at least β|F| of the sets of F. Theorem 1.1 was proved by Katchalski and Liu [18] and it is one of the key ingredients in the proof of the so-called Hadwiger-Debrunner ( p, q)-Conjecture [15] by Alon and Kleitman [4]. Definition 1. 2 We say that a family of convex sets has the ( p, q)-property, for p ≥ q, if for any p-subset F ∈ F p there is a q-subset F ∈ F q with non-empty common intersection F = ∅. [4]) For any d ≥ 1 and p ≥ q ≥ d + 1 there is a number P = P( p, q, d) with the following property: any finite family F of convex sets in R d with the ( p, q)-property can be pierced by P points.

Theorem 1.3 (The ( p, q)-theorem
The proof of Theorem 1.3 combines Theorem 1.1 with the following result of independent interest. Theorem 1.4 (Weak -net for points [1]) For any dimension d ≥ 1 and > 0 there is W = W ( , d) with the following property: for every finite (multi-)set P of points in R d one can find W points in R d that pierce every convex set A ⊆ R d with |A ∩ P| ≥ |P|.
Understanding the asymptotic behaviour of W ( , d) is one of the most challenging open problems in discrete geometry [19,Chap. 10]; see [9,10,20] for the best known bounds.
The starting point of our investigation is the Colorful Helly Theorem of László Lovász, first stated in [7], which concerns the scenario in which the intersecting (d +1)tuples form a complete (d + 1)-partite hypergraph. Definition 1. 5 We say that a finite family of convex sets F is k-colored if each set K ∈ F is colored with (at least) one of k distinct colors. The k-coloring of F can be expressed by writing F as a union of k color classes F 1 ∪ F 2 ∪ · · · ∪ F k , where each class F i consists of the sets with color i ∈ [k]. We say that the k-colored family F, with color classes F 1 , . . . , F k , has the Colorful Helly property, or CH( Notice that Theorem 1.6 says nothing about transversals to the remaining d color classes F j , with j ∈ [d + 1] \ {i}. The primary goal of this paper is to gain a deeper understanding of the transversals to all of the color classes F i in a (d + 1)-colored family F that satisfies CH. Theorem 1.6 is in close relation, via point-hyperplane duality, with the colorful version of the Carathéodory theorem due to Bárány [7]. Holmsen et al. [17] and independently Arocha et al. [6] recently established the following strengthening of Bárány's result: Theorem 1.7 (Very Colorful Carathéodory Theorem) Let P be a finite set of points in R d colored with d + 1 colors. If every (d + 1)-colorful subset of P is separated from the origin, then there exist two colors such that the subset of all points of these colors is separated from the origin.
Unfortunately, there is no Very Colorful Helly Theorem which guarantees that a second color class can be pierced with few points, as is illustrated by the following example (see Fig. 1). Let F d+1 = {R d } and, for each 1 ≤ i ≤ d let F i be a collection of hyperplanes orthogonal to the x i -axis. Then F d+1 is the only class that has a point transversal, moreover, each of the remaining classes may need an arbitrarily large number of points in order to be pierced. Note, though, that one can cross all the sets of d+1 i=1 F i by a single line.

Our Results
Our main result suggests that, in a sense, the scenario in Fig. 1 is the only possible unless an additional color class can be pierced by few points. In other words, Theorem 1.10 characterizes the families of sets with the Colorful Helly Property up to their transversal structure by flats.
This paper is organized as follows. In Sect. 2 we prove our main technical results-Theorems 1.9 and 1.10. To this end, we establish a series of claims of independent interest that concern 2-colored families of convex sets. Despite the apparent weakness of the 2-colored hypothesis in dimension higher than 2, these results provide all the essential ingredients for our analysis. Theorem 1.9 is finally established by repeatedly invoking a so-called "Step-Down" Lemma which provides a crucial relation between k-flat and (k − 1)-flat transversals of families with the Colorful Helly Property, for all The proof of the "Step-Down" Lemma is deferred to Sect. 3, and it is based on a careful adaptation of the machinery of Alon and Kleitman [4] and Alon and Kalai [2], to families of convex sets whose intersection graph is complete bi-partite. Section 4 is devoted to constructing a lower bound for g in Theorem 1.9. Our example implies that, independently of the value given to Finally, in Sect. 5 we conclude the paper with several intriguing questions for future study.

Proofs of Theorems 1.9 and 1.10
A crucial ingredient of our proof is the following claim which concerns 2-colored families.

Lemma 2.1 Let A and B be families of convex sets in R d such that A ∩ B = ∅ for every A ∈ A and B ∈ B. Then either
One can establish Theorem 1.9 in dimension d = 2 (with f (2) = 1 and g (2) ≤ 4) by applying Lemma 2.1 twice. The weaker transversal guarantee of Lemma 2.1 in higher dimension d ≥ 3 (namely, crossing by few hyperplanes instead of few lines) is due to the weaker, 2-colored hypothesis.
Proof Assume that (1) does not hold. Then by Helly's theorem there are convex sets Hence, the set B ∈ B must cross at least one of the respective bounding lines 1 and 2 of H 1 and H 2 to meet the sets A 1 , A 2 and A 3 Indeed, consider the arrangement of 1 , . . . , d and suppose that a set B ∈ B does not intersect any of the hyperplanes i . Then B must be completely contained in an open cell σ of their arrangement. Since B intersects each of the sets Both Theorems 1.9 and 1.10 are established by iterating the following more refined variant of Lemma 2.1.

Lemma 2.2 ("
Step-Down" Lemma) For any 1 ≤ k ≤ d and m ≥ 1 there exist numbers F(m, k, d) and G(m, k, d) with the following property.
Let A and B be finite families of convex sets in R d such that the family can be crossed by m k-flats. Then one of the following conditions is satisfied: Notice that the hypothesis of Lemma 2.2 implies, in particular, that every two sets A ∈ A, B ∈ B intersect. Thus, Lemma 2.1 deals with the special case of Lemma 2.
We defer the somewhat complex proof of Lemma 2.2 to Sect. 3. It combines the standard duality relation between transversal and packing numbers of hypergraphs with a "hyperplane" variant of Theorem 1.4, due to Alon and Kalai [2], in which we are given a collection of hyperplanes H and seek to find a small hyperplane transversal to all the convex sets that are crossed by a fixed fraction of the hyperplanes of H .
We are now ready to establish Theorem 1.9.
Proof of Theorem 1. 9 Let F be a d-colored family that satisfies CH(F 1 , . . . , F d ) and does not satisfy conclusion 1. Since the labeling of the color classes F 1 , . . . , F d is arbitrary, it suffices to show that the last family F d can be crossed by few lines. The underlying idea of our analysis is as follows. We apply the "Step-Down" Step-Down" Lemma can be used to further reduce the intrinsic "transversal dimension" of the remaining sets F i+1 , . . . , For reasons that will become evident shortly, we set Assuming neither of the families This proves Theorem 1.9 with

Remark 2.3
In the proof of Theorem 1.9, the value of g (d) can be further improved to by observing that at least one of the families F 1 , . . . , F d can be crossed by a single line. To this end, we project F in a generic direction ν and apply Theorem 1.6 to the resulting d-colored family This yields an intersecting color class F i ( ν) within R d−1 and, therefore, a ν-parallel line which crosses the respective color class F i .
Proof of Theorem 1. 10 The theorem is obviously true for d = 1 (with f (1, 1) = 1, g(1, 1) = 1). Assume with no loss of generality that the last color class F d+1 can be pierced by a point (in accordance with Theorem 1.6). We adopt the notation of the previous proof while dealing with the remaining color classes F 1 , . . . , F d .
Let l be the size of the largest sequence j 1 , j 2 , . . . , j l such that no class F j i can be pierced by F (M(l − 1, d), d − l + 2, d) points. Let F be the relabeling of F whose first l color classes satisfy F i = F j i , for 1 ≤ i ≤ l. By following the first l − 1 iterations of the proof of Theorem 1.9, we obtain that F l = F j l can be crossed by G(M(l − 1, d), d − l + 2, d) (d − l + 1)-flats. By reordering of j 1 , . . . , j l , this establishes the claim of Theorem 1.10 for F with where the second equality also uses that one of the color classes admits a transversal by a single point due to Theorem 1.6.

Proof of the "Step-Down" Lemma
We develop a bi-partite variant of the machinery that was used by Alon and Kleitman [4] to establish the ( p, q)-Conjecture (Theorem 1.3). This method was extended by Alon and Kalai [2] to obtain an analogous result for hyperplane transversals.

From Piercing to Packing Numbers
The crucial ingredient of Alon-Kleitman approach was a duality relation between transversal (or piercing), and packing (or matching) numbers of hypergraphs. for every x ∈ V. The fractional packing number ν * (G) of G is the total "weight" S∈E g(S) of the "heaviest" fractional packing g of G (that is, it is the largest possible value S∈E g(S) that can be attained by a fractional packing g).
A standard use of Linear Programming duality [2][3][4] yields the following relation between transversal and packing numbers of G.

Theorem 3.2 We have
for every hypergraph G and b ≥ 1.
The proof of Theorem 1.3 by Alon and Kleitman [4] combines the following key elements: • An abstract hypergraph G 0 (F), whose edges correspond to the sets of F, is constructed. Each vertex of G 0 (F) is a point that pierces some sub-family F ⊂ F. (To keep the vertex set finite, we have one vertex for each F ⊂ F with non-empty intersection F = ∅.) • The fractional packing number ν * (G 0 (F)) = τ * (G 0 (F)) is bounded from above using a suitable fractional Helly-type result (Theorem 1.1). • The fractional transversal for G 0 (F) is converted to an integral one using a weak -net result for point transversals [1].

Overview
As we cast the 2-colored setup of the "Step-Down" Lemma into the above abstract framework, several fundamental challenges are to be addressed. As we seek a relation between the transversal numbers of A and B, we maintain two hypergraphs G 0 (A) and G k−1 (B), where the former (resp., latter) hypergraph describes partial point (resp., (k − 1)-flat) transversals to A (resp., B). To show that at least one of G 0 (A) and G k−1 (B) has a bounded fractional packing number, we need a suitable fractional Helly-type result which is conveniently provided by the fractional variant of our 2-colored Lemma 2.1. Finally, to convert a fractional transversal for G k−1 (B) into an integral one, we need a small-size weak -net construction for (k − 1)-flats.
Unfortunately, no Helly-type results and no weak -net constructions are known for transversals by general (k − 1)-flats in R d , unless k = 1 [4] or k = d [2]. Note though that, in the scenario of Lemma 2.2, the pairwise intersections I(A, B) are assumed to "occur" within few k-dimensional flats of R d . We can therefore invoke the fractional variant of Lemma 2.1 in dimension k and similarly apply the weak -net construction of Alon and Kalai [2] for hyperplanes in R k .

Bounding the Fractional Packing Number
Let A and B be families of convex sets that satisfy the hypothesis of Lemma 2.2. That is, the family I(A, B) of pairwise intersections can be crossed by m k-flats 1 , . . . , m . Hypergraphs G 0 (A) and G k−1 (B) Below we define the abstract hypergraphs G 0 (A) and G k−1 (B) which describe, respectively, partial point transversals to A, and partial transversals by (k − 1)-flats to B.

The
The hypergraph G 0 (A) = (V A , E A ) is constructed analogously to the one of Alon and Kleitman [4]: for every subfamily A ⊂ A with A = ∅ we add a point x A ∈ A to V A , and for every convex set A ∈ F we add the edge To show that at least one of the hypergraphs G 0 (A) or G k−1 (B) has a bounded fractional packing number, we use the following fractional variant of our 2-colored Lemma 2.1. where the function β(·, ·) is defined as in the Fractional Helly Theorem 1.1. The reasons behind this choice will become evident during the proof. We may assume |A| ≥ 6d α , for otherwise γ |A| ≤ 1 and the result follows immediately.
For a subset A ∈ A d+1 , and B ∈ B, we say that (A , B) is a special pair if B intersects every set in A ; in other words, A and B form a star in the bipartite graph that represents pairwise intersections between the elements of A and B.
Let T denote the set of all the special pairs (A , B) as above. We first establish a lower bound for the cardinality of T . To this end, we claim that there are at least α 2 |B| heavy elements B of B each of which intersects with at least α 2 |A| elements of A. Indeed, otherwise we contradict the hypothesis as the number of pairwise intersections would be fewer than Let a = α 2 |A| and b = α 2 |B| . The discussion above shows that there are at least b heavy elements, each of which appears in at least a d+1 special pairs. Therefore: The second inequality is obtained as follows, where we use |A| ≥ 6d α at the end: Now, consider the subdivision T = T 1 T 2 : If at least dλ |A| d+1 of the (d + 1)-subfamilies of A are intersecting, then by the Fractional Helly Theorem 1.1 we obtain an intersecting subfamily of A of size γ |A| and we are done. Therefore, we may assume that less than dλ |A| d+1 of the (d + 1)subfamilies of A are intersecting. Since each of them appears in at most |B| special pairs of T 1 , we obtain Equations (3.1) and (3.2) imply that |T 2 | ≥ dλ|B| |A| d+1 . By the pigeon-hole principle there is a non-intersecting (d + 1)-subfamily A 0 ⊂ A that appears in at least dλ|B| special pairs. Let B 0 be the family of all the elements B in B which yield such a special pair (A 0 , B). Applying Lemma 2.1 to A 0 and B 0 we get a collection of d hyperplanes that cross all the sets in B 0 . Therefore, again by the pigeon-hole principle, one of these hyperplanes crosses at least 1 d |B 0 | ≥ λ|B| of the sets of B. Now we prove the following auxiliary statement. Claim 3. 4 We have that either ν * (G 0 (A)) ≤ 1/(γ (1/m, k)) or ν * (G k−1 (B)) ≤ 1/(λ (1/m, k)), where m, G 0 (A) and G k−1 (B) are as defined above, and the functions γ and λ are defined as in Lemma 3.3.

Proof
The fractional packing numbers ν * (G 0 (A)) and ν * (G k−1 (B)) exist as we are optimizing linear functions over a compact domain. Furthermore, the standard theory of Linear Programming implies that these values may be obtained via a pair of non-negative rational assignments f : E A → Q and g : E B → Q. 1 Given the contrapositive assumption, the following inequalities hold for all x 0 ∈ V A and σ 0 ∈ V B : .

Wrap-up
Combining Claim 3.4 with Theorem 3.2, we obtain that at least one of the graphs G 0 (A) and G k−1 (B) has a bounded fractional transversal number, so one of the following inequalities must hold: (1/m, k) , τ * (G k−1 (B)) ≤ 1 λ(1/m, k) .
Analogously to the proof of Claim 3.4, we obtain respectively either a rational (and not everywhere zero) function f : V A → Q + such that every edge e ∈ E A (representing some set A ∈ A) contains vertices (i.e., points) of total weight or a similar function g : V B → Q + such that every edge e ∈ E B contains vertices of total weight σ ∈e Similar to the proof of Claim 3.4, the rational assignments f and g yield either (i) a multiset of pointsV A ⊂ R d such that any member A of A contains at least γ (1/m, k)|V A | of these points, or (ii) a multisetV B of (k − 1)-flats within m i=1 i such that any member B of B is crossed by at least λ(1/m, k)|V B | of the flats.
In the former case, we use Theorem 1.4 to show that, in case (i), the family A can be pierced by points.
In the remaining case (ii), we use the following analogue of Theorem 1.4 for hyperplane transversals, due to Alon and Kalai [2]: We prove the result in the following two subsections. We begin with the case d = 2 which is later used to deal with the general case.

The Planar Construction
Let m = 2 f and T 0 be a triangle in the plane such that its bottom side is parallel to the x-axis. We first construct m triangles T 1 , . . . , T m , each with one horizontal side and vertices in the relative interiors of the three sides of T 0 , and such that no three of these triangles T i , T j , T k for 1 ≤ i < j < k ≤ m have a common intersection. A way to do this is to construct them recursively: we start with two arbitrary such triangles T 1 and T 2 and at each step i > 2 we place the horizontal side of T i sufficiently close to the horizonal side of T 0 so that it avoids all previous pairwise intersections (see Fig.  3). Let the first color class F 1 be the resulting family {T 1 , . . . , T m }. Clearly we need at least m/2 = f points to pierce F 1 .
Let E 1 , E 2 , E 3 be the three sides of T 0 . As each set of F 1 intersects the relative interior of each E i , for 1 ≤ i ≤ 3, we can slightly shrink each E i away from its adjacent vertices of T 0 while preserving the intersection with every element of F 1 . The family F 2 will consist of m slightly translated copies of each (previously shrunk) segment E i so that they still intersect every triangle in F 1 but are still pairwise disjoint. Note that we need at least 3m > f points to pierce F 2 .
In order to cross F 1 ∪ F 2 with lines, we need in particular to cross the interiors of E 1 , E 2 , E 3 , so at least two lines are needed.   2 are depicted. We have C 1 = conv(τ, v 4 ) and C 2 = conv(λ, v 1 ), with τ ∈ T 1 and λ ∈ T 2 . The sets ofF (3) 3 are the facets of the bounding simplex (3) vertices in the relative interiors of the three sides of τ i , such that no three of them intersect. LetF

Lemma 4.2 For any selection of C i ∈F
where relint(C) denotes the relative interior of C.
Proof We proceed by induction on the dimension d. For d = 2 we define the families in a similar way as for d > 2. Then, when d = 2, the colored familyF (2) is essentially the same as in the planar case, where by definition each triangle inF 1 intersects the relative interiors of the sides of T 0 τ 1 , which are precisely the elements ofF 2 .
Now assume that d > 2 and the statement is true in dimension d − 1. Note that the cross-sections ofF In order to show that d−1 i=1 C i intersects the relative interior of C d , we distinguish between three cases. In each case, we use the induction hypothesis to pick a pair of points d−2 i=1 C i on different faces of C d which span an open segment s in the relative interior of C d . We then use the definition of C d−1 to argue that it must intersect s. See   . . . , v d−2 , z}). By continuity of the barycentric coordinates, it is easy to verify that C must separate x and y within the (d − 1)-simplex C d . Hence, C d−1 ⊃ C must intersect the segment s = x y in its interior, which is a point in the relative interior of C d .
(2) The cases C d = conv(V \ {v d }) and C d = conv(V \ {v d−1 }) are analogous, so we may assume that we are in the former case. By the induction hypothesis we know that there is a point of d−2 i=1 C i in the relative interior of conv({v 1 , . . . , v d−2 , v d−1 }), say x. Therefore d−2 i=1 C i also contains the segment xv d+1 , which is contained in the face C d . We claim that C d−1 must intersect the interior of this segment. Indeed, let y be the point of . . . , v d−2 , y}). This set must intersect the segment s = xv d+1 as desired (for it separates x from v d+1 within C d ). Therefore C d−1 intersects s = xv d+1 in its relative interior as before.
We are almost done with the construction. In view of Lemma 4.2, we may shrink all the elements ofF (d) away from the (d − 2)-dimensional faces of (d) in such a way that they remain convex and the colorful intersections continue to be non-empty. In this way we obtain the families F we take an additional step: we take m parallel copies of each so that they still intersect every element of F 1 ∪ · · · ∪ F d−1 but are pairwise disjoint.
By the cut-off procedure, no three sets of the same F i intersect for i ∈ [d − 1] (as any such intersection would project to a triple intersection within T i ). Thus, in order to pierce any such F i at least m 2 = f points are needed. To cross F = F 1 ∪ · · · ∪ F d by lines we also need to cross the relative interiors of the facets of . No line can pierce more than two such interiors. Therefore, at least d+1 2 lines are needed. This concludes the proof of Theorem 4.1.

Discussion
We studied families of convex sets which satisfy the Colorful Helly hypothesis. Our Theorems 1.8 and 1.10 offer complementary relations between the "transversal dimensions" of individual color classes.
We conjecture that an even stronger phenomenon happens: It is easy to check that Conjecture 5.1 is sharp for families of flats. The most elementary instance of the conjecture arises for d = 3 and F 3 = {R 3 }. The remaining two classes F 1 and F 2 satisfy a 2-colored hypothesis. If one of the classes has a transversal by few points, then Conjecture 5.1 holds for the families, as the other class can simply be pierced by R 3 . Otherwise, by Lemma 2.1 both F 1 and F 2 can be pierced by few planes. Then the validity of Conjecture 5.1 in this case depends on the answer to the following question: Problem 5.2 Is it true that for any two families A, B of convex sets in R 3 such that A ∩ B = ∅ holds for all A ∈ A and B ∈ B, one of the families A or B can be crossed by O (1)

lines?
Another intriguing question is what are the "true" values of f (d) and g (d) for Theorem 1.9 or, more precisely, what is the relation between these parameters? For example, does the theorem still hold with f (d) = 1 and large enough g (d), as it happens for d = 2?