Research Statement. Andrew Suk

Transcription

1 Research Statement Andrew Suk 1 Introduction My research interests are combinatorial methods in discrete geometry. In particular, I am interested in extremal problems on geometric objects. My research can be summarized as follows: Extremal results for topological graphs. It follows from Euler s polyhedral formula that every planar graph on n vertices has at most O(n) edges. A natural question to ask is how much can we relax the condition of planarity without violating the conclusion? In joint work with Ackerman, Fox, and Pach [2], we studied several generalizations of planarity. In particular, we obtained new upper bounds on the maximum number of edges in a topological graph not containing a fixed size grid. Tangencies between convex bodies. I have studied the problem on bounding the number of tangencies in a family of n Jordan regions in the plane, which is related to questions on incidences and union complexity. Analyzing the structure of the union of convex bodies or other geometric objects in the plane and in higher dimensions is a classical topic in discrete and computational geometry, with many applications in motion planning and computer graphics (see [3], for a survey). In joint work with Pach and Treml [15], we obtained a tight bound on the maximum number of tangencies in a family of n convex bodies, which can be decomposed into k subfamilies of pairwise disjoint sets. Erdős-Szekeres-type theorems for monotone paths and convex bodies. A classic theorem of Erdős and Szekeres states that every sufficiently large point set in the plane in general position contains n members in convex position. I have studied the problem on generalizing the Erdős-Szekeres theorem to convex bodies in the plane. In joint work with Hubard, Montejano, and Mora [13], we gave the necessary and sufficient conditions for which noncrossing convex bodies in the plane are in convex position. As a corollary, we improved the previous known bounds on the Erdős-Szekeres theorem in the context of convex bodies. In joint work with Fox, Pach, and Sudakov [8], we further improved these bounds and established new Ramsey-type results for monotone paths in ordered hypergraphs. In what follows I will describe each of these areas and discuss my future research goals. 2 Extremal type results 2.1 Topological graphs A topological graph is a graph drawn in the plane with vertices represented as points and edges drawn by non-self-intersecting curves connecting the corresponding points. We say that a topological graph is simple if any pair of its edges have at most one point in common. Given a simple topological graph G = (V, E), edges e 1, e 2 E(G) cross if they have a point in common (other than their 1

2 endpoints), and they are said to be disjoint if they have no points in common. In joint work with Ackerman, Fox, and Pach [2], we obtained several extremal results on topological graphs not containing a natural (k, l)-grid, that is, a set of k pairwise disjoint edges that all cross another set of l pairwise disjoint edges. Theorem 2.1. (Ackerman, Fox, Pach, Suk) Every n-vertex simple topological graph with no natural (2, 1)-grid has at most O(n) edges. Theorem 2.2. (Ackerman, Fox, Pach, Suk) Every n-vertex simple topological graph with no natural (k, k)-grid has at most O(k 2 n log 4k 6 n) edges. As a corollary to Theorem 2.1, all simple topological graphs not containing a natural (2, 1)-grid has bounded chromatic number and a linear size independent set. Future research. I intend to continue studying extremal problems in topological graph theory. One of the main tools used in the proof of Theorem 2.2 is the following theorem from [9]. Theorem 2.3. (Fox, Pach, Tóth) Let C be a collection of n curves in the plane such that each pair of curves intersect at most once. If there are ɛn 2 pairs of curves that intersect, then there exists two subfamilies S 1, S 2 C of curves such that S 1, S 2 δn and every curve in S 1 intersects every curve in S 2 where δ depends only on ɛ. Hence if G has many crossing edges, then one can use Theorem 2.3 to conclude that there are two large sets of edges that all cross each other. Therefore the main difficulty in bounding the number of edges in a simple topological graph with no natural (k, k)-grid (or no k pairwise crossing edges), is the case when there are very few crossing edges. To handle this case, I intend to study the following conjecture. Conjecture 2.4. Let C be a collection of n curves in the plane such that each pair of curves intersect at most once. If there are ɛn 2 pairs of curves that are disjoint, then there exists two subfamilies S 1, S 2 C of curves such that S 1, S 2 δn and every curve in S 1 is disjoint to every curve in S 2 where δ depends only on ɛ. Note that if two curves can cross an arbitrary number of times, there is a simple construction of n curves in the plane with n 2 /8 disjoint pairs, but does not contain two large subfamilies disjoint from each other. If conjecture 2.4 were true, then it would be a powerful tool in proving many extremal problems in topological graph theory. 2.2 Tangencies between convex bodies Two convex sets in the plane are tangent to each other if they have precisely one point in common and their interiors are disjoint. It is was first observed by Tamaki and Tokuyama [22] that in order to obtain an upper bound on the number of incidences between a family of curves C and a set of points in the plane, one must compute the number of tangencies between the members of C. I have studied the structure of tangencies between two families of closed Jordan regions, each consisting of n pairwise disjoint members. It was shown by Pinchasi and Ben-Dan [6], who independently raised the same question, that the maximum number of such tangencies is O(n 3/2 log n). Their proof is based on a theorem of Marcus-Tardos [14] and Pinchasi-Radoičić [19]. They suspected that the correct order of magnitude of the maximum is linear in n. In joint work with Pach and Treml [15], 2

3 we show that this is not true even for x-monotone curves. However we do show that it is linear in n when both families consist of convex regions (convex bodies). Theorem 2.5. (Pach, Suk, Treml) The number of tangencies between two families of convex bodies in the plane, each consisting of n > 2 disjoint members, cannot exceed 8n 16. Let C be a family of n convex sets in the plane. Suppose that C can be decomposed into k subfamilies, each consisting of pairwise disjoint bodies. Applying Theorem 2.5 to each pair of these subfamilies separately and adding up the relevant bounds, we obtain the following corollary. Corollary 2.6. Let C be a family of n convex bodies in the plane, which can be decomposed into k subfamilies consisting of disjoint regions. Then the total number of tangencies between members of C is O(kn). This bound is tight up to a multiplicative constant. Future research. In [15], we conjectured the following. Conjecture 2.7. For every fixed integer k > 2, the number of tangencies in any n-member family of convex bodies, no k of which are pairwise intersecting, is at most O k (n). Very recently Ackerman settled the conjecture for k = 3 [1], by slightly modifying the proof of Theorem 2.5. I intend to extend his discharging technique to settle the case when k = 4. For larger values of k, I plan to study the problem on bounding the chromatic number of an intersection graph of convex sets in the plane. Given a collection of convex sets C, the intersection graph G(C) is a graph with vertex set C and two vertices are adjacent if their corresponding sets intersect. It has been conjectured that the chromatic number of such graphs is bounded by some function of its clique number. 3 Ramsey type problems 3.1 Convex bodies and ordered hypergraphs The classic 1935 paper of Erdős and Szekeres [7] published in Compositio Mathematica was one of the original results that let to a very rich discipline within combinatorics: Ramsey theory (see, e.g., [12]). The main result of their paper states the following. Theorem 3.1. For any positive integer n, let ES(n) denote the smallest number such that any set of ES(n) points in the plane such that no three are collinear contains n members in convex position. Then ( ) 2n 4 2 n ES(n) + 1 = O(4 n / n). n 2 Erdős and Szekeres conjectured that ES(n) = 2 n for every integer n. Currently the best known upper bound on ES(n) is ( ) 2n 5 n 2 +1 obtained by Tóth and Valtr [23]. Bisztriczky and Fejes- Tóth [4] later generalized the Erdős-Szekeres theorem to families of convex bodies in the plane [5]. A family C of convex bodies (compact convex sets) in the plane is said to be in convex position if none of its members is contained in the convex hull of the union of the others. We say that C is in general position if every three members of C are in convex position. 3

4 Bisztriczky and Fejes-Tóth showed that there exists a function D(n) such that every family of D(n) pairwise disjoint convex sets in general position contains n members in convex position. In [17], the Bisztriczky-Fejes Tóth theorem was extended to families of noncrossing convex bodies, that is, to convex bodies, no pair of which share more than two boundary points. It was proved that there exists a function N = N(n) such that from every family of N noncrossing convex bodies in general position in the plane one can select n members in convex position. In joint work with Hubard, Montejano, and Mora [13], we generalized the definition of order type to a family of convex bodies. Given a family C of noncrossing convex sets, we say that the ordered triple (A, B, C) C has a clockwise (counterclockwise) orientation if there are three points a A, b B, c C on the boundary of conv(a B C) that follow each other in clockwise (counterclockwise) order. Note that a triple (A, B, C) may have both orientations. See Figure 1. B B B A C A C C A (a) (A, B, C) with a clockwise orientation. (b) (A, B, C) with a counterclockwise orientation. Figure 1. (c) (A, B, C) with both a clockwise and a counterclockwise orientation. The order type of C is the mapping assigning each triple (A, B, C) C the orientation of that triple. The order type of a point set was introduced by Goodman and Pollack [10] in the early eighties, and has played a significant role in geometric transversal theory [24]. The main result in our paper was the following. Theorem 3.2. (Hubard, Montejano, Mora, Suk) A family of noncrossing convex bodies is in convex position if and only if there exists an ordering of the family such that every triple is oriented clockwise. As a corollary, we improved the previous known bounds on D(n) and N(n). ( (2n 5 ) ) (2n 4 ) Corollary 3.3. (Hubard, Montejano, Mora, Suk) D(n) + 1 n Corollary 3.4. (Hubard, Montejano, Mora, Suk) There exists a constant c such that N(n) 2 2cn. Inspired by Theorem 3.2, Hubard and Montejano asked if the order type of every family of convex bodies in the plane can be represented by a point set. This was answered in the negative by Pach and Tóth in the following Theorem Theorem 3.5. (Pach and Tóth [18] and Suk [20]) Let r(n) denote the largest integer such that every family C of n pairwise disjoint segments in the plane in general position has r(n) members whose order type can be represented by points. Then there exists an absolute constant c such that cn 1/4 < r(n) < n log 8/ log 9. n 2 4

5 In joint work with Fox, Pach, and Sukakov [8], we further improved the bound on N(n). Theorem 3.6. (Fox, Pach, Sudakov, Suk) N(n) 2 cn2 log n, where c is an absolute constant. One of the main tools we used in the proof of Theorem 3.6 was a Ramsey-type result on ordered hypergraphs. Let K k (N) denote the complete k-uniform hypergraph, consisting of all k-element subsets (k-tuples) of the vertex set [N] = {1, 2,..., N}. For any n positive integers, j 1 < j 2 < < j n, we say that the hyperedges {j i, j i+1,..., j i+k 1 }, i = 1, 2,..., n k + 1, form a monotone (tight) path of length n. Let N k (q, n) denote the smallest integer N such that for every q-coloring of the hyperedges (k-tuples) of K k (N), there exists a monotone path of length n such that all of its hyperedges are of the same color. Using this notation for graphs (k = 2), Dilworth s theorem can be rephrased as N 2 (q, n) = (n 1) q +1. For 3-uniform hypergraphs, a straightforward generalization of the cup-cap argument of Erdős and Szekeres [7] gives N 3 (2, n) = ( ) 2n 4 n For more (but fixed number) of colors, in [8] we proved that lower and upper bounds are tight apart from a logarithmic factor in the exponent. Theorem 3.7. (Fox, Pach, Sudakov, Suk) We have 2 (n/q)q 1 N 3 (q, n) n nq 1. Define the tower function t i (x) by t 1 (x) = x and t i+1 (x) = 2 t i(x). Using a stepping up approach, we also showed the following extension. Theorem 3.8. (Fox, Pach, Sudakov, Suk) For each k 3 and q there are constants c, c (depending only on k and q) such that t k 1 (cn q 1 ) N k (n, q) t k 1 (c n q 1 log n). As a geometric application, we proved that N(n) N 3 (3, n) 2 cn2 log n. Future research. I intend to continue studying the function N k (q, n). We plan to derive better upper bounds on N k (q, n) by modifying the proof of Theorem 3.8 and using more modern probabilistic methods. Furthermore, we plan to explore the other Ramsey-type functions described in [8] as well as developing density-type theorems for ordered hypergraphs. One of our main goals is to show that D(n) = N(n) = ES(n). 5

6 References [1] Ackerman, E.: personal communication, [2] Ackerman, E., Fox, J., Pach, J., and Suk. A.: On grids in topological graphs. In Proceedings of the 25th Annual Symposium on Computational Geometry (Aarhus, Denmark, June 08-10, 2009). SCG 09. ACM, New York, NY, , [3] Agarwal, P.K., Pach, J., and Sharir, M.: State of the union (of geometric objects), in: Surveys on Discrete and Computational Geometry, Contemp. Math. 453, Amer. Math. Soc., Providence, RI, 2008, [4] Bisztriczky, T., Fejes Tóth, G.: A generalization of the Erdős-Szekeres convex n-gon theorem, Journal für die reine und angewandte Mathematik 395 (1989), [5] Bisztriczky, T., Fejes Tóth, G.: Convexly independent sets, Combinatorica 10 (1990), [6] Ben-Dan, I. and Pinchasi, R.: personal communication, November [7] Erdős, P., Szekeres, G.: A combinatorial problem in geometry. Compositio Math. 2, (1935) [8] Fox, J., Pach, J., Sudakov, B., Suk, A.: Erdős-Szekeres-type theorems for monotone paths and convex bodies, manuscript, [9] Fox, J., Pach, J., and Tóth, Cs. D. : Intersection patterns of curves. Journal of the London Mathematical Society (to appear). [10] Goodman, J. E. and Pollack, R.: On the cambinatorial classification of nondegenerate configurations in the plane, J. Combinatorial Theory Ser. A 29 (1982) [11] Green, B., Tao, T.: The primes contain arbitrarily long arithmetic progressions. Annals of Math. 167 (2008), [12] Graham, R. L., Rothschild, B. L., Spencer, J. H.: Ramsey Theory, 2nd ed., Wiley, New York, [13] Hubard, A., Montejano, L., Mora, E., Suk, A.: Order types of convex bodies, Order, DOI: /s [14] Marcus, A. and Tardos, G.: Intersection reverse sequences and geometric applications, J. Combin. Theory Ser. A 113 (2006), [15] Pach, J., Suk, A., and Treml, M.: Tangencies between families of disjoint regions in the plane. In Proceedings of the 2010 Annual Symposium on Computational Geometry (Snowbird, Utah, USA, June 13-16, 2010). SoCG 10. ACM, New York, NY, , [16] Pach, J., Tóth, G.: A generalization of the Erdős-Szekeres theorem to disjoint convex sets, Discrete and Computational Geometry 19 (1998),

7 [17] Pach, J., Tóth, G.: Erdős-Szekeres-type theorems for segments and noncrossing convex sets, Geometriae Dedicata 81 (2000), [18] Pach, J., Tóth, G.: Families of convex sets not representable by points, Indian Statistical Institute Platinum Jubilee Commemorative Volume Architecture and Algorithms, World Scientific, Singapore, 2009, [19] Pinchasi, R. and Radoičić, R.: On the number of edges in geometric graphs with no selfintersecting cycle of length 4, in: Towards a Theory of Geometric Graphs, Contemporary Mathematics (J. Pach, ed.), 342, American Mathematical Society, Providence, RI, [20] Suk, A.: On the order types of system of segments in the plane, Order 27 (2010), [21] Szemerédi, E.: On sets of integers containing no k elements in arithmetic progression, Acta Arithmetica 27 (1975), [22] Tamaki, H., Tokuyama, T.: How to cut pseudoparabolas into segments, Discrete Comput. Geom. 19 (1998), no. 2, [23] Tóth, G., Valtr, P.: The Erdős-Szekeres theorem, upper bounds and generalizations, Discrete and Computational Geometry - Papers from the MSRI Special Program (J. E. Goodman et al. eds.), MSRI Publications 52 Cambridge University Press, Cambridge (2005), [24] Wenger, R.: Progress in geometric transversal theory. In: Chazelle, B., Goodman, J. E., Pollack, R. (eds.) Advances in Discrete and Computational Geometry, Contemp. Math., 223, pp Amer. Math. Soc. (1999) 7