The unavoidable arrangements of pseudocircles
TTHE UNAVOIDABLE ARRANGEMENTS OF PSEUDOCIRCLES
CAROLINA MEDINA, JORGE RAM´IREZ-ALFONS´IN, AND GELASIO SALAZAR
Abstract.
It is known that cyclic arrangements are the only unavoidable simple arrange-ments of pseudolines: for each fixed m ≥
1, every sufficiently large simple arrangement ofpseudolines has a cyclic subarrangement of size m . In the same spirit, we show that thereare three unavoidable arrangements of pseudocircles. Introduction A pseudoline is a noncontractible simple closed curve in the projective plane R P . An arrangement of pseudolines is a set of pseudolines that cross each other exactly once. Twoarrangements of pseudolines are isomorphic if the cell complexes they induce in R P areisomorphic. An arrangement of pseudolines is simple if no three pseudolines have a commonpoint. A simple arrangement is cyclic if its pseudolines can be labelled 1 , , . . . , m , so thateach pseudoline i ∈ [ m ] intersects the pseudolines in { , , . . . , m }\{ i } in increasing order,as in Figure 1. We use A m to denote a cyclic arrangement of size m . Figure 1.
The cyclic arrangement A .In the spirit of [10], the following states that cyclic arrangements are the only unavoidable arrangements of pseudolines. Theorem 1 ([8, Proposition 3.4.7]) . For each fixed m ≥ , every sufficiently large simplearrangement of pseudolines has a subarrangement isomorphic to A m . We prove an analogue of Theorem 1 for arrangements of pseudocircles. A pseudocircle is asimple closed curve in the sphere S . We use Gr¨unbaum’s original notion that an arrangementof pseudocircles is a set of pseudocircles that pairwise intersect at exactly two points, at whichthey cross, and no three pseudocircles have a common point [5]. This notion is still adoptednowadays [9], and some more general notions are also used in the literature [4, 6].In Figure 2 we illustrate arrangements C , C , C , and it is clear how to generalize them to C m , C m , and C m , for any m ≥
1. These are the unavoidable arrangements of pseudocircles.
Mathematics Subject Classification.
Primary 52C30; Secondary 05C10, 52C40. a r X i v : . [ m a t h . C O ] A ug heorem 2. For each fixed m ≥ , every sufficiently large arrangement of pseudocircles hasa subarrangement isomorphic to C m , C m , or C m . Figure 2.
The arrangements C (left), C (center), and C (right).We note that Theorem 2 is best possible, as no pseudocircle can be added to the collection { C m , C m , C m } . This follows since for all integers m, n with m ≤ n , all subarrangements of C n (respectively, C n , C n ) of size m are isomorphic to C m (respectively, C m , C m ). Thus itis accurate to say that these are the unavoidable arrangements of pseudocircles.For the rest of the paper, for brevity we refer to an arrangement of pseudocircles simplyas an arrangement .2. Reducing Theorem 2 to two kinds of arrangements
There are, up to isomorphism, only two arrangements of size 3. Following [3], these are the
Krupp arrangement and the
NonKrupp arrangement . We refer the reader to Figure 3. Notethat the Krupp arrangement is isomorphic to C , and the NonKrupp arrangement is iso-morphic to C and C . If an arrangement P of size at least 3 has all its 3-subarrangementsisomorphic to the Krupp arrangement (respectively, to the NonKrupp arrangement), thenwe say that P is Krupp-packed (respectively,
NonKrupp-packed ). Figure 3.
The Krupp arrangement (left) and the NonKrupp arrangement (right).We now state Theorem 2 for Krupp-packed and for NonKrupp-packed arrangements. Aswe shall see shortly, the general version of Theorem 2 easily follows as a consequence.
Lemma 3.
Theorem 2 holds for Krupp-packed arrangements.
Lemma 4.
Theorem 2 holds for NonKrupp-packed arrangements.
We use Ramsey theory in our arguments. We recall that the order of a hypergraph isits number of vertices, and r k ( (cid:96) , (cid:96) , . . . , (cid:96) n ) denotes the Ramsey number for complete k -uniform hypergraphs. That is, if each k -edge of a complete k -uniform hypergraph of orderat least r k ( (cid:96) , (cid:96) , . . . , (cid:96) n ) has colour i for some i ∈ [ n ], then there is an i ∈ [ n ] and a completesubhypergraph of order (cid:96) i , all of whose k -edges have colour i . roof of Theorem 2, assuming Lemmas 3 and 4. Let m ≥ p such that every Krupp-packedor NonKrupp-packed arrangement of size at least p has a subarrangement isomorphic to C m , C m , or C m . Let Q be an arrangement of size q = r ( p, p ). Regard Q as a complete3-uniform hypergraph, and colour a 3-edge blue (respectively, red) if the pseudocircles inthe 3-edge form an arrangement isomorphic to the Krupp arrangement (respectively, to theNonKrupp arrangement).Since q = r ( p, p ) it follows from Ramsey’s theorem that Q has a subarrangement P ofsize p that is either Krupp-packed or NonKrupp-packed. The assumption on p implies that P , and hence Q , contains a subarrangement isomorphic to C m , C m , or C m . (cid:3) We finish this section by proving Lemma 3. The rest of the paper is devoted to the proofof Lemma 4.
Proof of Lemma 3.
The key fact we use here is the following. Every Krupp-packed arrange-ment P (known in the literature as an arrangement of great pseudocircles ) can be obtainedfrom a simple arrangement L of pseudolines by suitably gluing together two wiring diagramsof L , as illustrated in Figure 4 (see [4, Section 3.2] and [12, Section 6.1.4]). For a discussionon the wiring diagram representation of a pseudoline arrangement we refer the reader to [2].If L is a cyclic arrangement A m , it is readily seen that P is isomorphic to C m . Figure 4.
Obtaining C by suitably gluing together two wiring diagrams of A .Let m ≥ P be a Krupp-packed arrangement of size p := r ( m, m ).Let L be a simple arrangement of pseudolines that induces P , in the sense of the previousparagraph. Since p = r ( m, m ), then by [11, Proposition 1.4] L has a cyclic arrangement A m as a subarrangement. The subarrangement of P induced by the pseudolines in A m isobtained by suitably gluing together two copies of A m , and so it is isomorphic to C m . (cid:3) Intersection codes
In the proof of Lemma 4 we use intersection codes, as developed in [7, 9]. This framework,in which one naturally encodes combinatorially essential information of an arrangement, canbe seen as a generalization of the axiomatization of oriented matroids based on hyperlinesequences [1], and its essence goes back to the work of Gauss on planar curves in the 1830s.Let P = { , . . . , n } be an arrangement. For each i ∈ P choose a point p i not containedin any other pseudocircle, and also choose one of the two possible orientations for i , so thatfor each pseudocircle we can naturally speak of a left side and a right side. Suppose thatas we traverse i starting at p i following the chosen orientation, we intersect j from the left(respectively, right) side of j . We record this by writing j + (respectively, j − ). By keeping rack of the order in which the intersections occur, we obtain the code of i in P . Thus thecode of each i is a permutation of (cid:83) j ∈ [ n ] \{ i } { j + , j − } . If we omit the superscripts + and − ,we obtain the unsigned code of i .For instance, suppose that P is C in Figure 2. Choose the counterclockwise orientationfor each pseudocircle. If we choose as starting point for 3 its leftmost point, then its code is1 + + − − + + − − , and its unsigned code is 12455421.We make essential use of the following. Proposition 5 ([7, Section 3],[9, Section 2]) . Let P , Q be arrangements, both of which havetheir pseudocircles labelled , , . . . , n , where an orientation and an initial traversal point hasbeen chosen for each pseudocircle in P and each pseudocircle in Q . Suppose that for each i ∈ [ n ] , the code of i in P is the same as the code of i in Q . Then P and Q are isomorphic. We now introduce some notation to describe the codes of C m and C m in a compact manner.Let i, m be integers such that 1 ≤ i ≤ m . We use [1 + : i + ) to denote the string 1 + + · · · ( i − + .In a similar spirit, we use ( i − :1 − ] to denote ( i − − ( i − − · · · − ; we use ( i − : m − ] to de-note ( i + 1) − ( i + 2) − · · · m − ; and we use [ m + : i + ) to denote m + · · · ( i + 2) + ( i + 1) + . Finally,we use [1 + − : i + i − ) to denote 1 + − + − · · · ( i − + ( i − − . Note that [1 + :1 + ) , (1 − :1 − ], and[1 + − :1 + − ) are empty strings, and ( m − : m − ] and [ m + : m + ) are also empty strings. Withthis notation, we have the following. Observation 6.
Label C m with the natural extension of the labelling of C in Figure 2.Orient all pseudocircles counterclockwise, and for each pseudocircle choose as initial traversalpoint its leftmost point. Then the code of each i in C m is [1 + : i + )( i − : m − ][ m + : i + )( i − :1 − ] . Observation 7.
Label C m with the natural extension of the labelling of C in Figure 2.Orient all pseudocircles clockwise, and for each pseudocircle choose as initial traversal pointits bottom right corner. Then the code of each i in C m is [1 + − : i + i − )( i − : m − ][ m + : i + ) . We close this section with a remark on NonKrupp-packed arrangements. We say that anarrangement is bad if its pseudocircles can be labelled 1 , . . . , n so that for each pseudocircle i = 1 , , . . . , n −
2, the unsigned code of i in the subarrangement { i, i + 1 , . . . , n } is ( i +1)( i + 1)( i + 2)( i + 2) · · · nn . Up to isomorphism, there is only one bad NonKrupp-packedarrangement of size 4, namely the arrangement X shown in Figure 5. This is easily checkedby hand, or by an inspection of [4, Figure 2], which contains all arrangements of size 4. Figure 5.
The arrangement X .In a bad NonKrupp-packed arrangement of size 5, all 4-subarrangements would then beisomorphic to X . A routine case analysis by hand shows that no arrangement of size 5satisfies this property. We highlight this remark, as we use it in the proof of Lemma 4. Observation 8.
There is no bad NonKrupp-packed arrangement of size (or larger). . Proof of Lemma 4
First we identify a property shared by C m and C m . To motivate this, we refer the readerto C and C , shown in Figure 2. If we perform the relabelling i (cid:55)→ i − , , , S \{ } are pairwisedisjoint, and appear in a rainbow-like fashion: the unsigned code of 0 is 12344321. We saythat an arrangement is rainbow if its pseudocircles can be labelled 0 , , . . . , n so that (I) oneof the components of S \ { } contains no intersections among the pseudocircles 1 , , . . . , n ;and (II) the unsigned code of pseudocircle 0 is 12 · · · nn · · · Proposition 9.
For each fixed integer n ≥ , every sufficiently large NonKrupp-packedarrangement has a rainbow subarrangement of size n . Proposition 10.
For each fixed integer m ≥ , every sufficiently large NonKrupp-packedrainbow arrangement contains a subarrangement isomorphic to C m or C m . Before proving these propositions, for completeness we give the proof of Lemma 4.
Proof of Lemma 4, assuming Propositions 9 and 10.
Obviously it suffices to prove Theorem 2for every integer m ≥
5. Let m ≥ n := n ( m ) such that every NonKrupp-packed rainbow arrangement contains a subarrange-ment isomorphic to C m or C m . By Proposition 9, there is an integer q := q ( n ) such thatevery NonKrupp-packed arrangement has a rainbow subarrangement of size n . Thus everyNonKrupp-packed arrangement of size at least q contains a subarrangement isomorphic to C m or C m . (cid:3) Proof of Proposition 9.
Let n ≥ p := r ( n, n )+1 and q := r ( p, p, p, p ).We let Q = { , . . . , q } be a NonKrupp-packed arrangement, and show that Q contains arainbow subarrangement of size n .Choose an arbitrary starting traversal point and orientation for each pseudocircle in Q .The NonKrupp-packedness of Q implies that if j, k, (cid:96) are pseudocircles in Q such that j < k < (cid:96) , then the unsigned code of j in the subarrangement { j, k, (cid:96) } is either (i) kk(cid:96)(cid:96) ; or(ii) (cid:96)(cid:96)kk ; or (iii) k(cid:96)(cid:96)k ; or (iv) (cid:96)kk(cid:96) .Regard Q as a complete 3-uniform hypergraph, and assign to each 3-edge { j, k, (cid:96) } with j 2, the unsignedcode of i (cid:48) in the subarrangement { i (cid:48) , ( i + 1) (cid:48) , . . . , p (cid:48) } is ( i + 1) (cid:48) ( i + 1) (cid:48) · · · p (cid:48) p (cid:48) . Thus P is abad arrangement of size p > 5, contradicting Observation 8. Thus not all 3-edges of P canbe of colour (i). An analogous argument shows that not all 3-edges can be of colour (ii).If all 3-edges of P are of colour (iv) then by relabelling the pseudocircles in the reverseorder we obtain an arrangement in which all 3-edges are of colour (iii). Thus we may assumethat all 3-edges of P are of colour (iii). In particular, the unsigned code of 1 (cid:48) in { (cid:48) , . . . , p (cid:48) } is 2 (cid:48) (cid:48) · · · p (cid:48) p (cid:48) · · · (cid:48) (cid:48) . Using that p − r ( n, n ), an application of Ramsey’s theorem showsthat there exist i (cid:48) , i (cid:48) , . . . , i (cid:48) n , with 2 (cid:48) ≤ i (cid:48) < i (cid:48) · · · < i (cid:48) n ≤ p (cid:48) such that one of the connectedcomponents of S \ { (cid:48) } contains no intersections among the pseudocircles i (cid:48) , . . . , i (cid:48) n . Thearrangement { (cid:48) , i (cid:48) , . . . , i (cid:48) n } is rainbow. To see this, it suffices to relabel 1 (cid:48) with 0, and i (cid:48) j with j , for each j = 1 , . . . , n . (cid:3) roof of Proposition 10. Let m ≥ q = r ( m, m, m ), and n = r ( q, q ).Let N = { , , , . . . , n } be a NonKrupp-packed rainbow arrangement. We show that N contains a subarrangement isomorphic to either C m or to C m .Our first goal is to show that we may assume that the layout of N is as shown on the righthand side of Figure 6. To achieve this, first we note that by performing a self-homeomorphismof the sphere we may assume that the pseudocircle 0 is the union of the Greenwich Meridianand the 180th Meridian, in particular passing through the north pole N and the south pole S . We orient 0 in the direction from S to N following the Greenwich Meridian, as on the lefthand side of Figure 6. NS N S k N k n k Figure 6. Initial setup in the proof of Proposition 10. On the left hand sidewe illustrate pseudocircle 0. On the right hand side we illustrate the easternhemisphere, which contains all intersections among the pseudocircles 1 , . . . , n .We illustrate the regions N k (grey) and S k (white) of pseudocircle k .Since N satisfies rainbowness Property (II), we may assume that as we traverse theGreenwich Meridian from S to N we intersect the pseudocircles 1 , , . . . , n in this order, and aswe traverse the 180th Meridian from N to S, we intersect them in the order n, . . . , , 1. Since N satisfies rainbowness Property (I), then either the eastern or the western hemisphere (saythe eastern one) contains all the intersections among the pseudocircles in N := { , , . . . , n } .We orient all pseudocircles in N so that as we traverse 0 along the Greenwich Meridian fromS to N the code we obtain is 1 − − · · · n − (that is, these pseudocircles hit 0 from its left handside). Thus as we traverse the 180th Meridian from N to S the code is n + ( n − + · · · + .Each pseudocircle k ∈ N decomposes the eastern hemisphere into two parts, a part N k thatcontains N, and a part S k that contains S . Thus the setup is as illustrated in Figure 6.Since N is NonKrupp-packed, for any three pseudocircles j, k, (cid:96) in N , either both inter-sections of j and (cid:96) occur in N k , or they both occur in S k . Regard N as a complete 3-uniformhypergraph, and assign to a 3-edge { j, k, (cid:96) } with j Illustration of the proof of Proposition 10.Regard Q as a complete 3-uniform hypergraph. We assign to a 3-edge { j, k, (cid:96) } with j To prove Theorem 2 we make repeated use of Ramsey’s theorem, and so an explicit boundfor this theorem, derived from our proofs, would be multiply exponential. With additionaleffort (and considerably more space) we can save several applications of Ramsey’s theorem,and show that for each fixed m ≥ 1, every arrangement of pseudocircles of size at least2 cm contains a subarrangement isomorphic to C m , C m , or C m . This bound is still doublyexponential in m . What is the best explicit bound that can be proved for this theorem? Acknowledgements We thank Stefan Felsner and Manfred Scheucher for making available to us their softwareto generate all arrangements of pseudocircles of small order; experimenting with this codewas very useful at the early stages of this project. The first author is supported by Fordecytgrant 265667. The second author is partially supported by PICS07848 grant. The thirdauthor is supported by Conacyt grant 222667 and by FRC-UASLP. References [1] J. Bokowski. Oriented matroids. In Handbook of convex geometry, Vol. A, B , pages 555–602. North-Holland, Amsterdam, 1993.[2] S. Felsner and J. Goodman. Pseudoline arrangements. 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