Tolerance in Helly Type Theorems

Tolerance in Helly Type Theorems L. Montejano ∗ and D. Oliveros† July 5, 2010

Abstract In this paper we introduce the notion of tolerance in connection with Helly type theorems and prove, using the Erd˝ os-Gallai theorem, that any Helly type theorem can be generalized by relaxing the assumptions and conclusion, allowing a bounded number of exceptional sets or points. In particular, we analyze some of the classical Helly type theorems, such as Caratheodory’s and Tverberg’s theorems, as well as some other interesting ones.

Key words.

1

Tolerance, Helly type theorems.

Introduction

Let U be a set and let P be any property, closed under inclusions, for subsets of U . Results

of the type “if every subset of cardinality µ of a finite family F ⊂ U has property P, then the

entire family F has property P” are called Helly type theorems. The minimum number µ for which the result is true is called the Helly number of the Helly type theorem (U , P, µ).

Helly’s theorem is perhaps one of the most widely used theorems in combinatorial geometry. It states that “a finite family of convex sets F in Rd has non-empty intersection if and only

if any subfamily of cardinality at most d + 1 has non-empty intersection.” Thus in Helly’s theorem, the “universe” U is the family of all convex sets in euclidean space Rd , the property P is to have a point in common, and the Helly number is d + 1. ∗ Supported † Supported

by CONACYT 41340 [email protected] by CONACYT CCDG 50151 [email protected]

1

One of the simplest Helly type results is when U is the set of points in some linear space,

the Helly property is “aligned,” and the Helly number is µ = 3. That is, if every three points are aligned, then all the points are aligned. Another interesting one is when U is the set of

points in the plane, the Helly number is µ = 4, and the property P is “in convex position.”

There exist many other relevant and classical Helly type theorems such as the Caratheodory, Kirchberger and Tverberg theorems, and many others from transversal theory. See [3], [7] for excellent references. In this paper we introduce the notion of tolerance in Helly type theorems through the following definition: Definition 1.1. We say that a family F ⊂ U has property P with tolerance k if there is G ⊂ F,

of cardinality k, such that F \ G has property P; in other words, if F has property P with the

possible exception of k of its elements.

One of the purposes of this paper is to prove in a purely combinatorial way the following theorem: Theorem 1.1 (Main Theorem). Let (U , P, µ) be a Helly type theorem. Given a positive integer t, there is a positive integer η(t) such that a family F ⊂ U has property P with tolerance t if

and only if every subfamily of F of cardinality at most η(t) has property P with tolerance t.

The best positive integer η(t) for which Theorem 1.1 is true depends clearly on the Helly type result in question. In Section 2, we use the Erd˝os-Gallai theorem from graph theory to establish our main result. Next, in Section 3, we discuss the tolerance Helly theorem. In Section 4, we discuss some other interesting tolerance Helly type theorems such as the tolerance versions of Radon’s theorem and Tverberg’s theorem, as well as possible generalizations of Larman’s theorem, together with a discussion of the best Helly number η(t) for some of these classical Helly type theorems.

2

Helly type theorems for covering numbers in hypergraphs

In this paper, a hypergraph or λ-hypergraph Gλ , λ ≥ 2, is a finite non-empty set of objects called vertices and denoted by V (Gλ ) together with a collection of subsets of V (Gλ ) of cardinality λ

called edges and denoted by E(Gλ ). In the literature such hypergraphs are known as λ-uniform hypergraphs. Note that G2 , where λ = 2, is a usual graph. 2

A vertex and an edge are said to cover each other in a λ-hypergraph Gλ if they are incident in Gλ . A vertex cover in Gλ is a set of vertices that covers all the edges of Gλ . The minimum cardinality of a vertex cover in λ-hypergraph Gλ is called the vertex covering number or transversal number of Gλ and is denoted by β(Gλ ). A λ-hypergraph Gλ , λ ≥ 2, is called

k-critical if β(Gλ ) = k and its transversal number decreases whenever an edge is deleted from E(Gλ ). The study of k-critical λ-hypergraphs was initiated with the 1961 paper of Erd˝os and Gallai [4] in which they proposed the following problem: Problem (Erd˝ os and Gallai [4]). Find the maximum number of vertices η(λ, k) that a k-critical, λ-hypergraph Gλ can have. In the same paper Erd˝os and Gallai proved various results, such as the following bounds, which will be useful throughout the present paper: 1. Every k-critical, 2-graph has at most 2k vertices. 2. Every 2-critical, λ-hypergraph can have at most #((λ + 2))/2)2 $ vertices. Later, in 1989, Z. Tuza [11] found sharp bounds for η(λ, k). Furthermore, Erd˝os and Gallai’s theorem can be restated as the following Helly type theorem for transversal numbers in hypergraphs. See [9] and [12] for two excellent surveys on the Erd˝os-Gallai theory. Throughout the rest of the paper, the bound η(λ, k) will be called the Erd˝os-Gallai bound. Theorem 2.1 (Erd˝os, Gallai). Let Gλ be a λ-hypergraph. Then β(Gλ ) ≤ k if and only if β(H λ ) ≤ k for every H λ subgraph of Gλ with |V (H λ )| ≤ η(λ, k + 1).

We now prove our main theorem, Theorem 1.1, and show that if η(t) is the smallest possible integer for which this theorem is true, then η(t) ≤ η(µ, t + 1). Proof. 1.1 Let (U , P, µ) be a Helly type result and let t be a positive integer. Let η(t) =

η(µ, t + 1), defined as in Theorem 2.1. Then for every F ⊂ U, let us define the µ-hypergraph

Gµ (F) such that the set of vertices V (Gµ (F)) = F and {α1 , . . . , αµ } ∈ E(Gµ (F)) if and only if the family {α1 , . . . , αµ } ⊂ F does not have the property P. Assume that every subfamily of F

of cardinality at most η(t) has property P with tolerance t. Then every subgraph H µ of Gµ (F)

with |V (H µ )| ≤ η(µ, t + 1) has the property that β(H µ ) ≤ t. Then by Theorem 2.1, β(Gµ ) ≤ t.

Furthermore, β(Gµ ) ≤ t implies that with the possible exception of t vertices, {a1 , . . . , at }, of 3

Gµ (F), all the others are independent; that is, V (Gµ (F)) \ {a1 , . . . , at } does not contain any edge of Gµ (F). This implies that every µ members of F \ {a1 , . . . , at } have property P and

therefore that all F \ {a1 , . . . , at } have property P. So F has property P with tolerance t. On the other hand, if a family F ⊂ U has property P with tolerance t, then any subfamily of F has property P with tolerance t.

Before we begin a discussion of our tolerance Theorem 1.1 in connection with the classical Helly type theorems and studying its Helly number η(t), first observe that the Erd˝os-Gallai Theorem 2.1 may be considered as a Helly type theorem with Helly number η(λ, k + 1). So, applying Theorem 1.1 to it and observing that the Erd˝os-Gallai theorem for k and tolerance t is the Erd˝os-Gallai theorem for k + t, we obtain the following inequality: η(λ, k + t + 1) ≤ η(η(λ, k + 1), t + 1). Also, note that as corollaries of Theorem 1.1, a wide variety of interesting results can be obtained. The three examples shown here are chosen at random, being no more important than many others which could have been presented instead. For instance, we know from [1] that if every six lines in a collection of lines in Rd have a transversal line, then the whole collection admits a transversal line. So, using the fact that η(6, 2) ≤ 16, we have the following: Corollary 2.1.1. If in a collection of lines in Rd , of every 16 lines, 15 of them have a transversal line, then there is a line transversal to all lines except one. Corollary 2.1.2. If in a collection of ordered plane convex sets, of every 81 convex sets, 78 of them have a transversal line consistent with the order, then with the possible exception of three, all of the convex sets have a transversal line. This follows from the Hadwigwer transversal theorem (see [7]) and the fact that η(3, 4) ≤

η(η(3, 3), 2) ≤ η(16, 2) ≤ 81. Finally,

Corollary 2.1.3. If in a collection of points in the plane, of every 30 points, 28 of them are in convex position, then all of them with the possible exception of two are in convex position. This follows from the fact that η(4, 3) ≤ η(η(4, 2), 2) ≤ η(9, 2) ≤ 30. Although Tuza’s bounds [11] for η(µ, k + 1) are the best possible for hypergraphs, in some cases, these bounds are far from being sharp in our case. Consider for example, the k-tolerance version of the aligned Helly result. 4

Proposition 2.1. Let F be a collection of at least 2t + 2 points. If every subfamily of F with

cardinality 2t + 2 is aligned with tolerance t, then all points of F, with the possible exception of t, are aligned.

Proof. Since every subfamily of F with 2t + 2 points is aligned with tolerance t, then there are t + 2 points, say {a1 , . . . , at+2 } of F in a line L. Then we may assume there are more than t points, say {b1 , . . . , bt+1 }, outside L. Consider the 2t + 2 points {a1 , . . . , at+1 , b1 , . . . , bt+1 }. By

the t-tolerance of F, t + 2 of such points must lie on a line L0 different from L, so L0 contains {b1 , . . . , bt+1 } and only one point of L, say a1 . But now, again by the t-tolerance, t + 2 of

the 2t + 2 points {a2 , . . . , at+2 , b1 , . . . , bt+1 } must lie along a line, contradicting the fact that L '= L0 .

Note that in this example, the best possible bound is η(t) = 2t+2 < η(3, t+1); in particular, when t = 1, our bound is η(1) = 4 and η(3, 2) = 6.

3

Tolerance Helly theorem for convex sets

In this section F will denote a finite family of convex sets in the d-dimensional euclidean space

Rd . Recall that a family F has a point in common with tolerance t if F has the property that

all members of the family with the possible exception of t of them, have a point in common. As a corollary of Theorem 1.1, we have the following tolerance Helly theorem.

Theorem 3.1. A family F of convex sets in Rd has a point in common with tolerance t if and

only if every subfamily F ! of F with |F ! | ≤ η(d + 1, t + 1) has a common point with tolerance t, where η(λ, t) is the Erd˝ os-Gallai bound.

When λ = 2, the Erd˝os-Gallai bounds give rise to sharp bounds in the case of the tolerance Helly theorem for the line. That is, if F is a family of intervals in R1 , then F has a point

in common with tolerance 1 if and only if every subfamily H, with |V (H)| ≤ 2t + 2, has a

common point with tolerance t. Furthermore, it is easy to see that this result is not true if |V (H)| ≤ 2t + 1. In the following theorem, we calculate the Helly number for the tolerance Helly theorem with tolerance 1. Theorem 3.2. Let F be a family of convex sets in Rd . Then F has a point in common with 5

tolerance 1 if and only if every subfamily H, with |V (H)| ≤# ((d + 3))/2)2 $, has a common point with tolerance 1. Furthermore, this bound is the best possible.

Proof. It is clear by Erd˝os-Gallai bound (2) that η(d + 1, 2) ≤ #(d + 3))/2)2 $. We shall prove that this theorem is not true if |V (H)| ≤ #((d + 3))/2)2 $ − 1. Fix a positive integer d ≥ 3. For every 2 ≤ n ≤ d − 1, we will construct a family F of

(n + 1)(d − n + 2) convex sets in Rd such that F does not have a common point with tolerance 1, but every subfamily of F of size (n + 1)(d − n + 2) − 1 has a common point with tolerance 1. Since the maximum of {(n + 1)(d − n + 2) | 2 ≤ n ≤ d − 1} = #((d + 3))/2)2 $, this will be enough to prove the theorem.

Given a set of points X = {x1 , . . . xm } in Rn , denote by cc(X) the convex hull of X and by

(X)i ⊂ X the subset {x1 , . . . x ˆi , . . . xm } of all the points in X with the exception of xi . Write Rd = Rn × Rd−n , where 2 ≤ n ≤ d − 1 and consider two simplexes ∆n+1 = cc({x1 , . . . xn+1 }) ⊂

Rn and ∆d−n+1 = cc({y1 , . . . yd−n+1 }) ⊂ Rd−n . Define

I = {x1 , . . . xn+1 } × {y1 , . . . yd−n+1 } and note that I is a finite subset of Rd with (n + 1)(d − n + 1) points. Next, we define a family F with (n + 1)(d − n + 2) convex sets in Rd . For i = 1, . . . , n + 1 and j = 1, . . . d − n + 1, let Ai := cc({x1 , . . . xn+1 }i ) × ∆d−n+1 and let

I (i,j) be the convex hull of all the points in I with the exception of the vertex (xi , yj ). By this we obtain a family of convex sets F = ∪n+1 Ai 1

!

1) Given i ∈ {1, . . . , n + 1}, we note that

"

∪i,j I (i,j) .

j#=i

Aj = {xi } × ∆d−n+1 and, since {xi } ×

∆d−n+1 ∩ I (i,1) ∩ I (i,2) · · · ∩ I (i,d−n+1) = ∅, we have that the family F \ Ai does not have

a point in common. 2) Similarly, since

"n+1 i=1

Ai = ∅, all the elements in F \ I i,j have empty intersection.

So, 1) and 2) yield that the family F does not have a point in common with tolerance 1.

Next, we prove that for every X ∈ F, the family F ! = F \ X has a point in common with

tolerance 1.

6

Assume first that X = Ai0 for some i0 ∈ {1, . . . n + 1}, so F ! = F \ Ai0 . Take j0 ∈

{1, . . . d − n + 1} and consider F ! \ I (i0 ,j0 ) . Note that (xi0 , xj0 ) ∈ I (i,j) , for every (i, j) '= (i0 , j0 ); " moreover (xi0 , xj0 ) ∈ {xi0 } × ∆d−n+1 = j#=i0 Aj . So F ! = F \ Ai0 has a point in common with tolerance 1.

Assume now that X = I (i0 ,j0 ) for some i0 ∈ {1, . . . n+1} and j0 ∈ {1, . . . d−n+1}. This time

the F ! = F \ I (i0 ,j0 ) have a point in common with tolerance 1, because again (xi0 , xj0 ) ∈ I (i,j) " for every (i, j) '= (i0 , j0 ) and (xi0 , xj0 ) ∈ {xi0 } × ∆d−n+1 = j#=i0 Aj . This completes the proof.

4

Tolerance in other Helly type theorems

4.1

Tolerance Caratheodory theorem

What is usually called Caratheodory’s theorem is a very simple statement that says that if A ⊂ Rd and x ∈ cc(A), then there is a subset B of A, of cardinality at most d + 1, such that

x ∈ cc(B). We now state a tolerance version of the Caratheodory theorem which is interesting in its own right.

Definition 4.1. Let F be a finite set of points in Rd and define the k-convex hull by cck (F) =

#

X⊂F , |X|=k

(F − X).

Thus cc0 (F) is the usual convex hull. !

Theorem 4.1. Let F be a finite set of points in Rd . Then x ∈ cck (F) if and only x ∈ cck (F ) !

!

for some F ⊂ F with | F |≤ η(d + 1, k + 1), where η(λ, k) is the Erd˝ os-Gallai bound.

Proof. It is well known that another statement of Caratheodory’s theorem is the following Helly type theorem. Let (Rd \ {0}, P, d + 1) be the Helly type theorem corresponding to the following property: A collection of points F ⊂ Rd \ {0} has property P if and only if F is

contained in an open half-space through the origin. By Theorem 1.1, the following is also true: A collection of points F ⊂ Rd \ {0} is contained in an open half-space through

the origin, with the possible exception of k of them, if and only if every subfamily 7

!

F ⊂ F of size η(d + 1, k + 1) is contained in an open half-space through the origin, with the possible exception of k of them. !

!

!

Thus 0 ∈ / cck (F) if and only if for every F ⊂ F, with | F |≤ η(d+1, k+1), we have 0 ∈ / cck (F ).

4.2

Tolerance Tverberg theorem

Let A and B be two collections of points in Rd . We say that A can not be separated from B even with a tolerance of k if for every subset X ⊂ A ∪ B, with |X| = k, A \ X can not be separated from B \ X. That is,

cc(A \ X) ∩ cc(B \ X) '= ∅. In 1972 David Larman [8] proved the following result: Let F be a set with 2d + 3 points

in Rd . Then there is a partition of F in two collections of points A and B such that for any x ∈ F , A \ {x} can not be separated from B \ {x}. Note that this theorem can be considered the tolerant version of the Radon theorem with tolerance 1, since the latter implies that A and B

can not be separated even with a tolerance of 1. This problem is closely related to the problem of how small a set of points must be in order to be projectively equivalent to the vertices of a convex polytope. Given integers d ≥ 1, k ≥ 0, let R(d, k) be the minimum positive integer such that given a

set F of R(d, k) points in Rd , there is a partition of F into two subsets of points that can not

be separated even with a tolerance of k. Then R(d, 0) is the Radon number d + 2. Furthermore,

R(d, 1) ≤ 2d + 3. The exact calculation of R(d, 1) is an interesting problem and the best

known lower bound is -5d/3. + 3 < R(d, 1), given by Ramirez-Alfonsin [10]. Furthermore, N. Garcia-Colin proved in [6] that R(d, k) ≤ (k + 1)d + k + 2.

We will say that a family F of τ subsets of points A1 , . . . , Aτ of Rd can not be separated

even with a tolerance of k if for every subset X ⊂ A1 ∪ · · · ∪ Aτ of cardinality k, cc(A1 \ X) ∩ · · · ∩ cc(Aτ \ X) '= φ.

Given integers d ≥ 1, τ ≥ 2, k ≥ 0, let T (d, τ, k) be the minimum positive integer n such

that given a set F of n points in Rd , there is a partition of F in τ subsets of points that can 8

not be separated even with a tolerance of k. Then T (d, τ, 0) = (τ − 1)(d + 1) + 1 is the classic Tverberg number and T (d, 2, k) = R(d, k).

Conjecture 4.2. Given integers d ≥ 1, τ ≥ 2, k ≥ 0, T (d, τ, k) ≤ (k + 1)(τ − 1)(d + 1) + 1. That is, given a set F of (k + 1)(τ − 1)(d + 1) + 1 points in Rd , there must be a partition of F into τ subcollections of points that can not be separated even with a tolerance of k.

4.3

Tolerance Convex Position theorem

This section is related to the following Helly type result:

If every four points of a finite set of points in the plane are in convex position, then all points are in convex position.

Theorem 1.1 implies that given a finite set F of points in the plane, if for every nine points

of F, eight are in convex position, then with the possible exception of one, all points of F are in

convex position. This follows from the fact that η(4, 2) = 9, where η(λ, k) is the Erd˝os-Gallai

bound. On the other hand, for every set of five points in the plane, fore of them are in convex position. This is the celebrated Esther Klein observation in the Erd˝os-Szekeres problem [5]. Furthermore, note that the vertices of a convex hexagon plus two more points in the interior of one of its diagonals, each one close to a vertex, give rise to an example of a finite set F

of points in the plane with the property that for every seven points of F, six are in convex

position, but it is not true that all points of F with the possible exception of one, are in convex

position. Then the exact bound should be between every nine or every eight. The purpose of the following theorem is to show that eight is the sharp bound. Theorem 4.3. Let F be a finite set of points in the plane. If for every eight points of F, seven

are in convex position, then with the possible exception of one, all points of F are in convex position.

Proof. Suppose the theorem is not true. Then there should be a counterexample F with nine

points. As usual, let G be the 4-hypergraph where the set of vertices is F and four points of

F give rise to an edge of G if and only if these four points are not in convex position. The hypergraph G has covering number β(G) = 2, but for every v ∈ F, β(G \ {v}) = 1; that is, 9

there is no convex octagon in F, but given any point v ∈ F, there is a point u ∈ F such that F \ {u, v} is a convex heptagon. From this, it is easy to see that any two edges of G intersect,

otherwise there is v ∈ F, with β(G \ {v}) '= 1. If every three edges of G have a vertex in

common, then all edges of G have a vertex in common, which is a contradiction to the fact that β(G) = 2. Then there are three edges, say e1 , e2 , e3 ∈ E(G), such that e1 ∩ e2 ∩ e3 = ∅. Assume

e1 ∩ e2 = {a}, e1 ∩ e3 = {b} and e2 ∩ e3 = {c}. Then e1 = {x1 , x2 , a, b}, e2 = {x3 , x4 , a, c}

and e3 = {x5 , x6 , b, c}, and then V (G) = {x1 , x2 , x3 , x4 , x5 , x6 , a, b, c}. Furthermore, it is not difficult to see that given any other edge e of E(G) different from e1 , e2 and e3 , {a, b, c} ⊂ e. Then if u ∈ {x1 , . . . , x6 }, there are at most two edges of G that contain u.

Let us consider the collection of points Υ = {x1 , x2 , x3 , x4 , x5 , b, c} ⊂ F ⊂ R2 . Every four

of them are in convex position because there is no edge of G contained in Υ ⊂ V (G). Thus the

points of Υ ⊂ R2 are in convex position. Let us denote by H the convex heptagon with vertices in Υ.

We shall analyze where the point x6 is. Assume first that the point x6 is in the interior of H. Then we want to prove that the point x6 must be in the 1-convex hull cc1 (Υ) (see Definition 4.1). Assume this is not the case. Then there is a point u ∈ Υ such that x6 ∈ / cc(Υ \ {u}); hence x6 is

in the triangle with vertices {v, u, w}, where {v, u, w} ⊂ Υ and v, u, w are consecutive vertices

in the heptagon H. The line through u and x6 cuts Υ in at most one more vertex z of H. Thus {x6 , v, u, w} is an edge of G and for every x ∈ Υ \ {u, v, w, z}, either {x6 , u, v, x} or {x6 , u, w, x}

is an edge of G. But this is a contradiction to the fact that there are at most two edges of G that contain x6 . This proves that x6 ∈ cc1 (Υ). Then by Theorem 4.1, there are six points {u1 , . . . , u6 } ⊂ Υ such that x6 ∈ cc1 ({u1 , . . . , u6 }).

Suppose without loss of generality that the order of u1 , . . . , u6 is consistent with the order in which these vertices appear in the convex hexagon cc({u1 , . . . , u6 }). Then {x6 , u1 , u3 , u5 } and

{x6 , u2 , u4 , u6 } are edges of G. Moreover, since the point x6 is in at most three diagonals of the convex heptagon H, there is a point y ∈ Υ with the property that the line through y and x6

does not cut the heptagon H at a vertex, which implies that there is a triangle with vertices in Υ different from {u1 , u3 , u5 } and {u2 , u4 , u6 } that contains the point x6 in its interior. This is again a contradiction to the fact that there are at most two edges of G that contain x6 . So the point x6 must lie outside the convex heptagon H. Let L1 and L2 be the two support lines of H through the point x6 , and let wi ∈ Li ∩ F

for i = 1, 2. Consider the triangle T with vertices {x6 , w1, w2 }. Then there must be at least

one point of F in the interior of T , otherwise there is a octagon with vertices in F, which is

10

impossible. On the other hand, there are at most two points of F in the interior of T , since there are at most two edges of G that contain x6 .

Suppose there is a point w ∈ Υ that lies in the interior of T . Then {x6 , w, w1 , w2 } is an

edge of G. Furthermore, since the points of Υ are in convex position, the line through w and x6 cuts Υ at at most one more vertex z ∈ H. Therefore for every x ∈ Υ − {w1 , w2 , w, z}, either

{x, w, w1 , x6 } or {x, w, w2 , x6 } is an edge of G. But this is a contradiction to the fact that there are at most two edges of G that contain x6 . Therefore no points of Υ lie in the interior of T .

This proves that x6 is not a point outside H, either. Hence the counterexample F with nine points does not exist.

4.4

Tolerance colorful theorem

Using Theorem 2.1 and following the spirit of the proof of Theorem 1.1, it is possible to obtain several tolerance colorful theorems in the sense of B´ar´any and Lov´asz (see [2]). As a sample, we include here the colorful tolerance Helly theorem in the plane with tolerance 1. Theorem 4.4. Let F be a finite set of convex sets in the plane painted with three colors. Suppose that for every subfamily F ! ⊂ F of cardinality 6, there is A ∈ F ! such that every heterochromatic

triple contained in F ! \ A is intersecting. Then there is a color with the property that all convex sets of F of this color, with the possible exception of one, have a point in common.

Proof. Let G be the following 3-hypergraph: the set of vertices V (G) = F and {A1 , A2 , A3 } ∈

E(G) if and only if the triple {A1 , A2 , A3 } ⊂ F is heterochromatic and non-intersecting. By

hypothesis, every subgraph H ⊂ G of size smaller or equal than 6 has vertex covering number β(H) = 1. By the Erd˝os-Gallai Theorem 2.1, and since η(3, 2) = 6, we have that β(G) = 1.

Thus there is a vertex A ∈ V (G) that covers all edges of the 3-hypergraph G. This implies that

F \ A is a family of convex sets in the plane painted with three colors and with the property that every heterochromatic triple is intersecting. Then by the Colorful Helly Theorem [2], there is a color with the property that all convex sets of F of this color, with the possible exception of A, have a point in common.

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