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Details of project

  
Identifier
60427
Type K
Principal investigator Bárány, Imre
Title in Hungarian Kombinatorikus módszerek a diszkrét geometriában
Title in English Combinatorial methods in discrete geometry
Keywords in Hungarian diszkrét geometria, konvexitás,
Keywords in English discrete geometry. convexity,
Discipline
Mathematics (Council of Physical Sciences)100 %
Panel Mathematics and Computing Science
Department or equivalent Alfréd Rényi Institute of Mathematics
Participants Füredi, Zoltán
Kincses, János
Pach, János
Pór, Attila
Solymosi, Jozsef
Tóth, Géza
Starting date 2006-02-01
Closing date 2010-06-30
Funding (in million HUF) 16.000
FTE (full time equivalent) 4.58
state closed project
Summary in Hungarian
Több területen fogunk dolgozni:

1. 0-1 politópok maximális lapszáma.
Maximálisan hány (n - 1)-dimenziós lapja lehet egy n-dimenziós 0-1 politópnak? Erre szeretnénk minél pontosabb választ adni.

2. Geometriai és topologikus gráfok.
Kialakulófélben van az ilyen gráfokra vonatkozó extremális gráfelmélet, Ramsey-elmélet, véletlen gráfelmélet: ezen a területen fogunk dolgozni.

3. Geometriai gráfok alkalmazásai.
A fenti módszereket két területen is alkalmazni szeretnénk: véges pontrendszerek távolságeloszlásának analízisére, és gráfrajzoló algoritmusok kidolgozására.

4. Pontrendszerek kevés távolsággal.
Egy nagy ponthalmaz sok távolságot határoz
meg, de hogy mennyit az a témakör egyik legszebb és legrégibb megoldatlan problémája. Új eredményeket kivánunk elérni általános Minkowski terekben.

5. Szögek eloszlása.
Erdőstől származik az a kérdés, hogy mennyi hegyesszög fordulhat elő n egysíkban lév½o független pont között? Itt rengeteg érdekes probléma vizsgálható.

6. Erdős-Szekeres típusú problémák.
Az Erdős-Szekeres tételnek számos általánosítása és módosítása ismert. Ezek közül szeretnénk többet is kiterjeszteni konvex halmazokra.

7. Véletlen politópok.
A véletlen politópok elmélete az elmúlt egy-két évben új lendületet kapott: sikerült a megfelel½o funkcionálok eloszlásáról értékes információt bizonyítani. Az ilyen irányú kutatásokba fogunk bekapcsolódni.

8. Helly dimenzió, VC-dimenzió.
Tervbe vettük a Hanner politópok Helly dimenziójának meghatározását. Tervezzük továbbá az adott Vapnik-Chervonenkis dimenziójú telített hipergráfok kombinatorikus struktúrájának vizsgálatát
Summary
We plan to work in several areas:

1. Maximal number of facets of 0-1 polytopes.
At most how many (n − 1)-dimensional facets can an n-dimensional 0-1 polytope have? We want to answer this question asymptotically.

2. Geometric and topological graphs.
There is an emerging theory of extremal graphs, Ramsey theory, and random graph theory for geometric and topological graphs: we plan to work in this area.

3. Applications of geometric graphs.
We hope to apply the new methods to attack two problems: one is the analysis of the distribution of distances among finitely many points, and the other is the design and analysis of graph-drawing algorithms.

4. Point sets with few distances.
A large point set determines many distances. But determining the minimal number of distinct distances is one of the oldest and one of the most beautiful unsolved problems in discrete geometry. We plan to prove new results on this problem in general Minkowski spaces.

5. Distribution of angles.
How many acute angles can occur between n general position points in the plane? This question is due to Erdős. There are loads of interesting questions in this area.

6. Erdős-Szekeres type problems.
The famous Erdős-Szekeres theorem has several generalisations and extensions. We plan to extend some of these results from point sets to convex sets.

7. Random polytopes.
In the last two years exciting new developments have taken place in the theory of random polytopes: quite recently, significant information has become available about the distribution of various functionals of random polytopes. We plan to join in in these investigations.

8. Helly dimension, VC-dimension.
We want to determine the Helly dimension of Hanner polytopes. We also plan to investigate the combinatorial structure of saturated hypergraphs with given Vapnik-Chervonenkis dimension.




 

Final report

  
Results in Hungarian
Barany Imre veletlen politopokkal, a Tverberg tetel altalanositasaival, a topologia es kombinatorikus geometria hatarteruletevel foglalkozott. Ugyanezen a teruleten dolgozott Kincses Janos, es meg politopok kombinatorikajan. Furedi Zoltan a ter illetve egy korlatos halmaz gazdasagos fedeseivel, grafok pakolasaival foglalkozott. Solymosi Jozsef additiv kombinatorikaval illetve additiv szamelmelettel foglalkozott, elsosorban azzal, hogy egy n elemu A szamhalmaz eseten legalabb mekkora az |A+A|+|A*A| ertek. Ezenkivul a Szemeredi-Trotter illetve Pach-Sharir tetelhez hasonlo incidencia tetelt bizonyotott magas dimenzios algebrai gorbekre. Ez utobbi eredmeny nagyon igeretes kezdetnek tunik. Toth Geza grafok metszesi szamaval es lerajzolasaival foglalkozott, konstualt egy grafot, amelynek a par-metszesi szama es paratlen-metszesi szama elter. Pach Janossal tanulmanyoztak grafok sikbeli es magasabb genuszu feluleten vett metszesi szamait, illetve az ezek kozti osszefuggeseket. A sik illetve a ter sokszoros fedeseinek szetbonthatosagat is vizsgalta. Pach Janossal azt vizsgaltak, hogy konvex halmazok rendtipusa mikor reprezentalhato pontokkal. Pach Janos Jacob Fox-szal a Lipton-Tarjan szeparator tetelt altalanositotta sikgrafokrol kulonbozo mas tipusu grafokra, peldaul konvex halmazok illetve gorbek metszetgrafjara. Por Attila lathatosagi grafokkal illetve grafok Kneser-reprezentaciojaval foglalkozott, amely szoros kapcsolatban van a frakcionalis kromatikus szammal.
Results in English
Imre Barany investigated random polytopes, generalizations of the Tverberg theorem, and problems on the boundary of combinatorial geometry and topology. Janos Kincses also worked in this latter area, and also studied combinatorics of polytopes. Zoltan Furedi studied economical coverings of the space, or a bounded set. He also obtained important results concerning packings of small graphs into a large graph. Jozsef Solymosi worked in additive combinatorics and additive number theory. He investigated especially the question that at least how large is |A+A|+|A*A| if A is a set of n numbers. He also proved an incidence result for high dimensional algebraic curves, similar to the Szemeredi-Trotter or the Pach-Sharir theorems. This result seems to be a good start. Geza Toth investigated crossing numbers and drawings of graphs. He constructed a graph whose pair-crossing number is larger than its odd-crossing number. With Janos Pach he studied relationships between crossing numbers of graphs on different surfaces. He also obtained results on the decomposability of multiple coverings of the plane or space. With Janos Pach he studied, under what conditions can the order type of convex sets be represented by points. Janos Pach, together with Jacob Fox, generalized the Lipton-Tarjan separator theorem for planar graphs, for intersection graphs of convex sets, and for intersection graphs of curves. Attila Por obtained results on visibility graphs and on Kneser representations of graphs.
Full text https://www.otka-palyazat.hu/download.php?type=zarobeszamolo&projektid=60427
Decision
Yes





 

List of publications

  
I. Barany, G Rote: Strictly convex drawings of planar graphs, Dokumenta Math, 11, 369-391, 2006
J. Pach and G. Tóth: How many ways can one draw a graph?, Combinatorica 26, 559--576, 2006
J. Pach and A. Dumitrescu: Pushing squares around, Graphs and Combinatorics 22, 37--50., 2006
J. Pach and D. Pálvölgyi: Bounded-degree graphs can have arbitrarily large slope numbers, Electronic J. Combinatorics 13, 4 pages., 2006
J. Montellano-Ballesteros, A. Pór, R. Strausz: Tverberg-type theorems for separoids, Discrete Comput. Geom., 35, 513--523, 2006
A. Fiat, J. Matouvsek, E. Mossel, J. Pach, M. Sharir, S Smorodinsky, U. Wagner, E. Welzl: Online conflict-free coloring for intervals, SIAM J. Computing 36, 956--973., 2006
J. Pach and G. Tardos: Forbidden paths and cycles in ordered graphs and matrices, Israel J. Math. 155, 309--334., 2006
I. Barany, J. Matousek: Berge's theorem, fractional Helly, and art galleries, European J. Comb. 36, 2303--2313, 2006
I. Barany, B. Doerr: Balanced partitions of vector sequences, Lin. Alg. Appl. 414, 464--469, 2006
I. Barany, M. Prodromou: On maximal convex lattice polygons inscribed in a plane convex set, Israel J. Math. 154, 337--360, 2006
I. Barany: Convex bodies, random polytopes, and approximation, Chapter in Stochastic Geometry, ed. W. Weil, Springer,, 2006
I. Barany: The probability that a convex body is lattice point free: a relative of Buffon's needle problem, Random Structures and Alg. 30, 414--426, 2007
I. Barany, J. Matousek: Quadratic lower bound for the number of colourful simplices, SIAM J. Discrete Math, 2007
Füredi, Z., Hwang, Kyung-Won, Weichsel, P. M.: A proof and generalizations of the Erdős-Ko-Rado theorem using the method of linearly independent polynomials, Topics in discrete mathematics, Algorithms Combin., 26,215--224, Springer,Berlin, 2006, 2006
Füredi, Z., Pikhurko, O, Simonovits M: 4-books of three pages. (English summary), J. Combin. Theory Ser. A 113 (2006), no. 5, 882--891, 2006
Füredi Z, Naor Assaf, Verstraete J: On the Turán number for the hexagon, Advances in Mathematics, 37, 476--496, 2007
Füredi Z, Katona, Gyula O. H.: 2-Bases of quadruples, Combinatorics, Probability and Computing, 15,1-2(2006),131--141, 2006
Füredi Z, Kündgen, A: Moments of graphs in monotone families, Journal of Graph Theory, 51, 37--48, 2006
J. Pach and G. Tóth: Crossing number of toroidal graphs,, in: Graph Drawing (P. Healy et al., eds.), Lecture Notes in Computer Science 3843, Springer, Berlin, 2006, 334--342., 2006
J. Pach, R. Radoicic, G. Tardos, and G. Tóth: Improving the Crossing Lemma by finding more crossings in sparse graphs, Discrete and Computational Geometry 36, 527--552, 2006
J. Pach, R. Radoicic, J. Vondrák: Nearly equal distances and Szemerédi's regularity lemma, Computational Geometry: Theory and Appls. 34, 11--19, 2006
J. Pach, R. Radoicic, J. Vondrák: On the diameter of separated point sets with many nearly equal distances, European J. Combinatorics 27, 1321--1332., 2006
J. Pach and G. Tóth: Comments on Fox News,, Geombinatorics 15, 150--154., 2006
J. Pach and G. Tóth: Degenerate crossing numbers, 22nd ACM Symposium on Computational Geometry, ACM Press,New York, 2006, 255--258., 2006
B. Keszegh, J. Pach, D. Pálvölgyi, G. Tóth: Drawing cubic graphs with at most five slopes, Graph Drawing 2006, Lecture Notes in Computer Scie114--125.nce 4372, Springer, 2007,, 2006
K. Böröczky, J. Pach and G. Tóth: Planar crossing numbers of graphs embeddable in another surface, Internat. J. of Foundations of Comp. Science 17, 1005--1011., 2006
J. Pach, G. Tóth: Crossing number of toroidal graphs,, in: Topics in Discrete Mathematics (M. Klazar et al., eds), Algorithms and Combinatorics 26, Springer, Berlin, 2006, 581--590, 2006
J. Pach, R. Pinchasi, M. Sharir: Solution of Scott's problem on the number of directions determined by a point set in 3-space, Discrete and Computational Geometry 38, 399--441, 2007
U. Adamy, M. Hoffmann, J. Solymosi, and M. Stojakovic: Coloring Octrees,, Theoretical Computer Science. 363, 11 - 17., 2006
Graham, Ron and Jozsef Solymosi: Monochromatic equilateral right triangles on the integer grid, Topics in Discrete Mathematics. Series: Algorithms and Combinatorics. Vol. 26. Ed. M Klazar, J. Kratochvil, M Loebl, J Matousek, R Thomas and P Valtr. Springer,, 2006
Jozsef Solymosi: Dense arrangements are locally very dense I, SIAM Journal on Discrete Mathematics 20, 623 - 627., 2006
Karolyi, Gyula and Jozsef Solymosi: ''Erdos-Szekeres theorem with forbidden order types'', Journal of Combinatorial Theory, Series A. 113, 455 - 465, 2006
Jozsef Solymosi: Arithmetic Progressions in Sets with Small Sumsets, Combinatorics, Probability and Computing. 15, 597 - 603, 2006
J.Solymosi and Cs. D. Toth: On distinct distances in homogeneous sets in the Euclidean space, Discrete and Computational Geometry 35, 629 - 634., 2006
Laba, Izabella and Jozsef Solymosi.: Incidence theorems for pseudoflats, Discrete & Computational Geometry, 2007
Jozsef Solymosi.: Elementary methods in Additive Combinatorics, Additive Combinatorics, AMS, 1-- 11 pages, 2007
Haart, Derrick, Alex Iosevich and Jozsef Solymosi.: Sum-product estimates in finite fields via Kloosterman sums, INTERNATIONAL MATHEMATICS RESEARCH NOTICES, 1--14, 2007
G. Tardos, G. Tóth: Decomposing multiple coverings with triangles, Discrete and Computational Geometry, 38, 443--450, 2007
G. Tóth: Note on the odd-crossing number and the pair-crossing number, Proceedings of the 19th Canadian Conference on Computational Geometry, Ottawa, Canada, 2007
J. Pach, G. Tóth: Decomposition of multiple coverings into many parts, Proceedings of the 23rd Annual Symposium on Computational Geometry, Gyeongju, South-Korea, 2007, 133-137, 2007
J. Cerny, J. Kyncl, G. Tóth: Improvement on the decay of crossing numbers, Proceedings of Graph Drawing 2007, Lecture Notes in Computer Science, 4732, 2007
A. Pór, D. Wood: Colourings of the cartesian product of graphs and multiplicative Sidon sets, Electronic J. Discrete Math. 28, 33-40, 2007
A. Pór, P. Valtr: On the Positive Fraction Erdős--Szekeres Theorem for Convex Sets, European Journal of Combinatorics, 27, 1199--1205, 2006
G. Tardos, G. Tóth: Crossing stars in topological graphs, SIAM J. on Discrete Math. (SIDMA) 21, 737-749, 2007
G. Tardos, G. Tóth: Decomposing multiple coverings with triangles, Discrete and Computational Geometry 38, 443-450, 2007
J. Pach, G. Tardos, G. Tóth: Indecomposable coverings, Discrete Geometry, Combinatorics and Graph Theory, Lecture Notes in Computer Science, 4381, 135--148, 2007
J. Pach, G. Tóth: Families of convex sets not representable by points, Architecture and Algorithms, World Scientific, Singapore, to appear, 2007
R. Radoicic, G. Tóth: The discharging method in combinatorial geometry and the Pach--Sharir conjecture, (J. E. Goodman, J. Pach, J. Pollack, eds.), Contemporary Mathematics, AMS, 453, 319-342., 2008
J. Pach, R. Pinchasi, M. Sharir: Solution of Scott's problem on the number of directions determined by a point set in 3-space, Discrete and Computational Geometry 38, 399--441, 2007
H. Bronnimann, J. Lenchner, J. Pach,: Opposite-quadrant depth in the plane, Graphs and Combinatorics 23, 145--152, 2007
E. Ezra, J. Pach, M. Sharir: On regular vertices of the union of planar objects, 23rd Symposium on Computational Geometry, ACM Press, New York, 220--226, 2007
J. Fox, J. Pach, Cs. D. Toth: A bipartite strengthening of the Crossing Lemma, Lecture Notes in Computer Science 4875, Springer-Verlag, Berlin, 13--24, 2008
J. Fox, J. Pach, Cs. D. Toth: Turan-type results for partial orders and intersection graphs of convex sets, Israel J. Math., accepted, 2007
J. Fox, J. Pach: A bipartite analogue of Dilworth's theorem for multiple partial orders, Eoropean J. Combinatorics, accepted, 2007
X. Chen, J. Pach M. Szegedy, and G. Tardos: Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles, Symp. on Discrete Algorithms (SODA 2008), ACM, New York--SIAM, Philadelphia, 94--101, 2008
N. Eaton, Z. Furedi, A. Kostochka, and J. Skokan: Tree representations of graphs, European Journal of Combinatorics 28, 1087--1098, 2007
D. Danev, Z. Furedi, and M. Ruszinko: Multiple access Euclidean channel, Multiple Access Channels - Theory and Practice, NATO Security through Science Series, 10, 54--72, 2007
I. Barany, V. V. Vu: Central limit theorems for Gaussian polytopes, Annals of Prob. 35, 1593--1621, 2007
I. Barany, J. Matousek: Packing cones and their negatives in space, Discrete Comp. Geom. 37, 177-187, 2007
J. Solymosi, G. Tardos: On the number of k-rich transformations, 23rd Symposium on Computational Geometry, ACM Press, New York, 1--9, 2007
I. Laba, J. Solymosi,: Incidence theorems for pseudoflats, Discrete and Computational Geometry, 37, 163--174, 2007
Chang, Mei-Chu, J. Solymosi,: Sum-product theorems and incidence geometry, J. European Math. Society, 9, 545--560, 2007
J. Kincses: The topological type of the alpha-sections of convex sets, Advances in Mathematics, 217, 2159-2169, 2008
G. Tardos, G. Tóth: Decomposing multiple coverings with triangles, Discrete and Computational Geometry, 38, 443--450, 2007
J. Kincl, J. Pach, G. Toth: Long alternating paths in bicolored point sets,, Special Volume of Discrete Mathematics Honouring the 60th birthday of M. Simonovits, 308, 4315-4322., 2008
G. Tóth: Note on the pair-crossing number and the odd-crossing number, Discrete and Computational Geometry, 39, 791-799., 2008
J. Pach, G. Tóth: Monochromatic empty triangles in two-colored point sets, Geometry, Games, Graphs and Education: the Joe Malkevitch Festschrift (S. Garfunkel, R. Nath, eds.), COMAP, Bedford, MA, 195-198, 2008
J. Pach, G. Tóth: Decomposition of multiple coverings into many parts, Computational Geometry: Theory and Applications, accepted, 2008
D. Palvolgyi, G. Tóth:: Convex polygons are cover-decomposable, Discrete and Computational Geometry, accepted, 2008
D. Kral, E. Macajova, A. Por, J.-S. Sereni :: Characterization results for Steiner triple systems and their application to edge-colorings of cubic graphs,, Canadian Journal of Mathematics, to appear, 2008
P. Hamburger, A. Por, M. Walsh:: Characterization results for Steiner triplPKneser representations of grape systems and their application to edge-colorings of cubic graphs,, SIAM J. on Discrete Math (SIDMA) accpeted, 2008
I. Barany, A. Por, P. Valtr:: Paths with no small angles,, LATIN 2008: Theoretical Informatics, Vol 4957, 654-, 2008
Z. Furedi and M. Ruszinko:: Large convex cones in hypercubes,, Discrete Applied Mathematics 156, 1536-1541, 2008
Z. Furedi, A. Gyarfas, G. N. Sarkozy, and S. Selkow:: Inequalities for the First-fit chromatic number,, Journal of Graph Theory 59 (2008), 75-88, 2008
Z. Furedi and J-H. Kang:: Covering the n-space by convex bodies and its chromatic number,, Discrete Mathematics 308 (2008), 4495-4500., 2008
Z. Furedi and L. Ozkahya:: On 14-cycle-free subgraphs of the hypercube., Combinatorics, Computing and Probability, accepted, 2008
I. Barany, A. Hubard, J. Jeronimo:: Slicing convex sets and measures by a hyperplane,, Discrete Comp. Geom., 39, 67--75,, 2008
I. Barany:: Extremal problems for convex lattice polytopes: a survey,, Contemporary Mathematics, 453, 87-103, Surveys on Discrete and Comp. Geometry, Ed.: J. E. Gooodman et al. AMS,, 2008
I. Barany:: Random points and lattice points in convex bodies,, Bulletin of the AMS, 45, 339-365, 2008
I. Barany:: On the power of linear dependencies,, Building Bridges, ed: M. Grotschel, G.O.H Katona, Springer, 2008, 31--46, 2008
G. Ambrus, I. Barany:: Longest convex chains, Random structures and Algorithms, 35, 137--162, 2009
J. Solymosi, K. Swanepoel:: Elementary incidence theorems for complex numbers and quaternions,, SIAM JOURNAL ON DISCRETE MATHEMATICS. 22: 1145, 2008
J. Solymosi:: Incidences and the Spectra of Graphs,, Building Bridges, ed: M. Grotschel, G.O.H Katona, Springer, 2008, 499-513, 2008
J. Solymosi, Van H. Vu:: Near optimal bound for the distinct distances problem in high dimensions,, COMBINATORICA. 28, 113 - 125., 2008
J. Solymosi, Cs. D. Toth:: On a question of Bourgain about geometric incidences,, Combinatorics, Probability and Computing. 17: 619 - 625, 2008
S. Bereg, A. Domitrescu, J. Pach,: Sliding disks in the plane, International J. on Computational Geometry and Appl. 18, 373--387., 2008
G. Calinescu, A. Domitrescu, J. Pach,: Reconfigurations in graphs and grids,, SIAM J. Discrete Mathematics 22, 124--138., 2008
F. Eisenbrand, T. Rothvoss, J. Pach, and N. Sopher:: Convexly independent subsets of the Minkowski sum of planar point sets, Electronic J. of Combinatorics 15 (2008), 2008
J. Fox, J. Pach:: Coloring $K_k$-free intersection graphs of geometric objects in the plane, Proc. 24th Symposium on Computational Geometry, ACM Press, New York, 2008, 346--354, 2008
R. Fulek, A. Holmsen, J. Pach:: Intersecting convex sets by rays,, Proceedings 24th annual Symposium on Computational Geometry, ACM Press, 385--391, 2008
J. Fox, J. Pach:: Erd\H os--Hajnal-type results on intersection patterns of geometric objects, Horizons of Combinatorics (E. Gy\H ori et al., eds.), Bolyai Soc. Math. St, 2008
P. Agarwal, J. Pach, M. Sharir:: State of the Union (of geometric objects),, Surveys on Discrete and Computational Geometry (J. E. Goodman et al., eds.), Contemporary Mathematics, vol. 453, AMS, Providence, RI, 9--48, 2008
J. Kincses: The Helly dimension of the L_1-sum of convex sets,, accepted in Acta Sci. Math., Szeged., 2010
D. P\'alv\"olgyi, G. T\'oth: Convex polygons are cover-decomposable, Discrete and Computational Geometry, 43, 483-496, 2010
A. Dumitrescu, J. Pach, G. T\'oth: Drawing Hamiltonian cycles with no large angles, Graph Drawing 2009, Lecture Notes in Computer Science , 5849, Springer-Verlag, Berlin, 2010
A. Dumitrescu, J. Pach, G. T\'oth: A note on blocking visibilities between points, Geombinatorics, 19, 67-73, 2009
J. Bar\'at, G. T\'oth: Towards the Albertson conjecture, Electronic Journal of Combinatorics, 17 (1), R73, 2010
P. Cheilaris, G. T\'oth: Graph unique-maximum and conflict-free colorings, Proceedings of the 7th International Conference on Algorithms and Complexity (CIAC), Lecture Notes in Computer Science, 6078, Springer-Verlag, Berlin, 143- 154, 2010
J. Pach, G. T\'oth: Families of convex sets not representable by points, Indian Statistical Institute Platinum Jubilee Commemorative Volume--Architecture and Algorithms, Vol. 3, World Scientific, Singapore, 43-53, 2009
Solymosi, Jozsef and Frank De Zeeuw: On a question of Erd?s and Ulam, Discrete Comp. Geometry, 43, 393-401, 2010
Solymosi, Jozsef: Spectral bounds on incidences, Additive Combinatorics. Ed. Javier Cilleruelo, et al. Advanced Courses in Mathematics CRM Barcelona. Basel: Birkhauser. 299-313, 2009
Pach, Janos, Jozsef Solymosi and Gabor Tardos: Crossing numbers of imbalanced graphs, JOURNAL OF GRAPH THEORY. Published Online: 11 Aug 2009 6 pages, 2009
Jozsef Solymosi and Van H Vu: Sum-product estimates for well-conditioned matrices, Bulletin of the London Mathematical Society, 41, 817-822, 2009
Jozsef Solymosi and Van H Vu: Some Ramsey results for the n-cube, J. Comb. Theory, Series A, 117, 189-195, 2010
Jozsef Solymosi and Van H Vu: Sumas contra productos?, Gaceta de la Real Sociedad Matematica Espanola, 12, 707-719, 2009
J. Arocha, I. Barany, X. Bracho, R. Fabilla, L. Montajano: Very Colourful theorems, Discrete Comp. Geom., 42, 142--154, 2009
I. Barany, P. Blagojevic, A. Szucs: Equipartitions by a convex 3-fan, Advances in Math., 223, 579--593, 2010
I. Barany, A. Por: Infinite paths with no small angle, Mathematika, 56, 35--44, 2010
I. Barany, A. Por, Valtr: Paths with no small angle, SIAM J. Discrete Math., 23, 1655--1666, 2009
R. Fulek and A. Holmsen. J. Pach: Intersecting convex sets by rays, Discrete and Computational Geometry 42, 343--358., 2009
R. Fulek and A. Holmsen. J. Pach. H.Tverberg: Intersecting convex sets by raysPints surrounding the origin, Combinatorica 28, 633-644., 2008
B. Keszegh, D. P\'alv\"olgyi, J. Pach and G. Toth: Cubic graphs have bounded slope parameter, J. Graph Algorithms and App lications 14, 5--17, 2010
J. Fox, J. Pach: A separator theorem for string graphs and its applicat ions, Proc. WALCOM: Algorithms and Computation, Lecture Notes in Computer Science 5431, Springer-Verlag, Berlin, 1--14, 2009
J. Pach, G. Tardos: Conflict-free colorings of graphs and hypergraphs, Combinatorics, Probability \& Computing 18, 819--834, 2009
E. Ezra, J. Pach, M. Sharir: On regular vertices of the union of planar objects, Discrete and Computational Geometry 41, 216--231, 2009
X. Chen, M. Szegedy, J.Pach, and G. Tardos: Delaunay graphs of point sets in the plane with respect to axis-parallel rectangles, Random Structures and Algorithms 34, 11--23, 2009
J. Pach, Tardos and G. T\'oth: Indecomposable coverings, Canadian Mathematical Bulletin 52, 451--463, 2009
J. Pach and G. T\'oth: Degenerate crossing numbers, Discrete and Computational Geometry 41, 376--384, 2009
J. Fox, J. Pach and G. T\'oth: A bipartite strengthening of the Crossing Lemma, Combinatorial Theory, Ser. B 100, 23--35, 2010
A. Dumitrescu, J. Pach, G. T\'oth: A note on blocking visibility between points, Geombinatorics, 19, 2009, 67--73., 2009
J. Fox, F. Frati, J. Pach and R. Pinchasi: Crossings between curves with many tangencies, Proc. WALCOM 2010: Workshop on Algorithms and Computation, Lecture Notes in Computer Science 5942, Springer-Verlag, 1--8., 2010
P. Brass, W. Moser and J. Pach: Research Problems in Discrete Geometry, Japanese translation: Springer, Tokyo, 2009, 2009
J. Pach and M. Sharir: Combinatorial Geometry and its Algorithmic Applications: The Alcala Lectures, Mathematical Surveys and Monographs, Vol. 152, AMS, Providence, 2009
Tobias Muller, A. Por, J-S. Sereni: Lower bounding the boundary of a graph in terms of its maximum or minimum degree, Discrete Mathematics, 308, 6581--6583, 2008
N. Lichiardopol, A. Por, J-S. Sereni: A step toward the Bermond?-Thomassen conjecture about disjoint cycles in digraphs, SIAM J. Discrete Math. 23, 979--992, 2009
P. Hamburger, A. Por, M. Walsh: Kneser representations of graphs, SIAM J. Discrete Math. 23, 1071--1081, 2009
A. Por, M. D. Woods: On Visibility and Blockers, to appear in Journal of Computational Geometry, 2010
M. Dunkum, P. Hamburger, A. Por: Destroying Cycles in Digraphs,, to appear in Combinatorica, 2010



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