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Borbély, József

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Institute of MathematicsResults of the Ph.D. thesisSome Ramsey- andanti-Ramsey-type results incombinatorial number theory andgeometryJózsef BorbélyDoctoral School: MathematicsDirector: Miklós Laczkovich, D.Sc.Professor, Member of the Hungarian Academy of SciencesDoctoral Program: Pure MathematicsDirector: András Sz¶cs, D.Sc.Professor, Corresp. Member of the Hungarian Academy of SciencesSupervisors: Gyula Károlyi, D.Sc.András Sárközy, Professor EmeritusEötvös Loránd University, Hungary, Budapest, 20151Summary of the thesisIn this thesis my goal is to present some applications of combinatorial methodsin number theory, group theory, and geometric graph theory. Combinatorialmethods are considered as profound and current topics of mathematics nowa-days. In particular, I will study Ramsey and anti-Ramsey (or rainbow type)problems. The great number of journals publishing papers in the concernedareas attests to the important role of these branches. The following jour-nals publish papers of this type (without making any claims of completeness):Combinatorica , Discrete and Computational Geometry, Integers, Journal ofCombinatorics and Number Theory, Moscow Journal of Combinatorics andNumber Theory, The Electronic Journal of Combinatorics. We always willconcentrate us on Ramsey-type problems. It is important to clarify what wemean by Ramsey-type results. In this thesis we speak about a Ramsey-typetheorem, if the result can be formulated in the following way: if one colors everyelement of a "large" structure with exactly one of k ﬁxed colors, then one canalways ﬁnd a monochromatic partial structure of a desired size (independent ofthe chosen coloring). The ﬁrst known theorem of this sort was proved by FrankP. Ramsey (see [13]). Since his death in 1930 many new Ramsey-type problemshave been posed and solved. This branch of mathematics has a widespreadliterature. The book of Graham, Rotschild and Spencer ([13]) is one of themost profound summaries of the most important results. In addition, the bookof Landman ([27]) contains many important results concerning Ramsey-typenumber theoretical problems. We have to distuingish between Ramsey-typetheorems and density theorems. We speak about a density theorem if we canformulate our results in the following way: if one chooses a "large" subset ofa given set, then one can ﬁnd another subset of the chosen subset with thedesired properties. It is clear that density theorems are stronger than Ramsey-type theorems. This means that it is possible, that one can ﬁnd a Ramsey-typeresult without even when no density result exists, but from the existence of adensity theorem one can conclude that there is also a Ramsey-type theorem.We have to explain what we understand by rainbow Ramsey-type theorems.2In these cases one colors every element of a given set by exactly one of thek given colors (in the same manner as in the case of Ramsey-type theorems),but now we look for multicolored substructures of given type. In this thesiswe focus primarily on Ramsey-type theorems in number theory and its relatedareas.The ﬁrst chapter is based on [2]. In this chapter we proceed from a problem ofRoth (see [17]) that was solved by Erdös, Sárközy and T. Sós in an elementaryway (see [11]). We will extend their methods in order to achieve more gen-eral results. For any k-coloring of a special subset of r-dimensional vectors weinvestigate asymptotically the minimal number of r-dimensional vectors thatare representable as a sum of distinct r-dimensional vectors of the same color.Throughout the ﬁrst chapter we use the following notations: for any integersk and l, where k ≤ l, (k, l) denotes the set {k, k + 1, · · · , l}. Moreover, for anypositive integer r and for any sets H1, H2, · · · , Hr, we denote the Cartesianproduct of these r sets by H1 ×H2 × · · · ×Hr. If H1 = H2 = · · · = Hr = H,then we write Hr = H1 × H2 × · · · × Hr. In Chapter 1 ﬁrst we prove thefollowing additive theorem:Theorem 1. For ﬁxed positive integers r, s, k there is a positive integer m0with the following property: For any positive integer m > m0 and for anyk-coloring of the elements of the set (1,m)r there are at least[ms]r− 3 · (2r−1mr)1−2−ks+k−1vectors ~x in (1,m)r, such that there are pairwise distinct vectors ~x1, ~x2, · · · , ~xsof the same color in (1,m)r, whose sum is ~x.Moreover, in the ﬁrst chapter we achieve a result concerning the number ofrepresentations:Theorem 2. For every positive real number α and β with the propertyαr + βr ≤ 122r+1k,there is a positive integer mαβ, such that for every m > mαβ and for everyk-coloring of Nr the number of elements in (1,m)r having representations as3a sum of two monochromatic distinct vectors in more than βr2mr ways is morethan αrmr.The second chapter is based on [3]. In the second chapter we approach theproblem of Roth from another point of view. This means that we look foranalogous results in some abelian groups, but this time we focus on rainbow-type Ramsey problems. More precisely, for any k-coloring of some specialabelian groups G we investigate the minimal number of elements that arerepresentable as a sum of r elements of distinct colors. First we prove thefollowing theorem in an elementary way:Theorem 3. For every k-coloring of the positive integers 1, 2, ..., n, wherecolor i occurs exactly ni times (i = 1, 2, ..., k), where n1 ≥ n2 ≥ · · · ≥ nk, thereare at most r− 1 +∑r−1i=1 (r − i)ni numbers in the set {1, 2, · · · , n} having norepresentation as a sum of r integers of distinct colors (r ≥ 1).Then we investigate analogous questions in ﬁnite abelian groups. We provethe following theorem by using Kneser's theorem:Theorem 4. For every k-coloring of the elements of the additive abelian groupG of ﬁnite order m, where every color class is used, there are at leastmin{m− (k − 1) ·mDk (G),mdk (G)}elements of the group having a representation as the sum of k elements of Gof diﬀerent colors (where dk (G) is the largest positive integer not greater thank dividing the number |G|, and Dk (G) is the smallest positive integer greaterthan k dividing the number |G|).Then, by using Vosper's theorem, we achieve the following result in Zp:Theorem 5. If the elements of Zp are colored by k ≥ 2 colors such that everycolor class is used, then at least p− 1 elements of Zp have a representation asa sum of k − 1 elements of pairwise distinct colors.Moreover, we have the conjecture, that even the following stronger resultholds:4Conjecture 1. If the elements of Zp are colored by k ≥ 4 colors such thatevery color class is used, then each element of Zp has a representation as asum of k − 1 elements of pairwise distinct colors.It is a natural question how to extend the result of Theorem 5 in other ﬁniteabelian groups. We could not solve the problem, but we have the following aconjecture:Conjecture 2. Let G be a ﬁnite abelian group having m elements, and let rand k be positive integers such that 2 ≤ r < k. If m = p1 · p2 · · · ps, wherep1 ≤ p2 ≤ · · · ≤ ps are prime numbers, then the following statement is true:for every k-coloring of the given abelian group G, where every color class isused, the number of elements of G having a representation as a sum of relements of distinct colors is(i) exactly m, if r = 2 and m < 2k − 1(ii) at least m− mp1, if r = 2 and 2k − 1 ≤ m(iii) at least m− mp1·p2···pk−r+1 , if 2 < r and k − r + 1 ≤ s(iv) is eactly m, if 2 < r and k − r + 1 > s.The bounds expressed above are tight.It is trivial that Conjecture 1 is a consequence of Conjecture 2.In the third chapter we generalize analogous Ramsey-type results concerningproducts. For any k-coloring of the positive integers up toM will estimate theminimal number of positive integers that are representable as a product of rintegers of the same color. The results of this chapter are written in [4]. Inthe third chapter we ﬁrst prove the following theorem:Theorem 6. There is a positive absolute constant c such that if r and kare positive integers, where k is greater than r, and M is a positive integerlarge enough depending on k, then every k-coloring of the elements of the set{1, 2, · · · ,M} satisﬁes the inequality∑b∈B1b> c · 2r· 1kr−1· logM,5where B denotes the set of positive integers that are representable as a productof r elements of the set {1, 2, · · · ,M} of the same color.Moreover, we investigate analogous problem in some ﬁnite abelian groups.We prove the following theorem:Theorem 7. For any positive integers k and r there exists a positive integerTk,r, such that for every positive integer M greater than Tk,r the followingstatement holds: for any k-coloring of the elements of any cyclic multiplicativegroup G of order M there are at leastM(M, r)− 3 ·(M(M, r))1−2−(k·(r−1)+1)elements of G that have a representation as a product of r distinct elements ofthe same color (here (M, r) denotes the greatest common divisor of the numbersM and r).In the fourth chapter we study the corresponding rainbow problems con-cerning products. We achieve an exact asymptotic result. More precisely, ifk is a ﬁxed positive integer and we color the positive integers up to n by kcolors so that every color is used, then by denoting the minimal number ofintegers having a representation as a product of r numbers of distinct colorsby R(r)k (n), we prove the following:Theorem 8. limn→∞R(r)k (n)n= 2− 12(r−1)(k−1)−(r−1)2.In the last chapter we present and solve a Ramsey-type problem in geomet-ric graph theory concerning multiplicity of special subgraphs. The problemwas posed by Gyula Károlyi (see [22]). A geometric graph is a graph drawnin the plane, where every vertex is represented by a point and the edges arestraight lines connecting some of the vertices (we assume, that an edge joiningtwo vertices does not pass through a third vertex). A convex geometric graphis a special geometric graph, where the vertices of the graph lie in a convexposition.The study of the Ramsey-multiplicity for abstract graphs was introduced in6the paper of Harary and Prins (see [19]). One can similarly introduce thisdeﬁnition for geometric graphs.For a graph G the geometric Ramsey number Rg (G) denotes the smallest in-teger r with the property that for every 2-coloring of the edges of a completegeometric graph H on at least r vertices, there is a subgraph of H with non-crossing edges, that is isomorphic to G and has all of its edges of the samecolor.One can similarly deﬁne the number Rc (G). In this case we restrict the deﬁ-nition to convex geometric graphs.The above numbers exist if and only if the graph G is outerplanar (see [14]).The concept of Ramsey multiplicity is the same as in the case of abstractgraphs. RMg (G) denotes the minimum number of monochromatic copies of agiven geometric graph G as subgraphs of the complete geometric graph H onRg (G) vertices, which is colored by 2 colors. One can similarly interprete thenumber RMc (G) in the case of convex geometric graphs.Let us denote the non-crossing matching with 2n vertices byM2n for geometricgraphs. In [24] the authors proved the following theorem for geometric graphs:Theorem 9. Rg (M2n) = Rc (M2n) = 3n− 1In [22] Károlyi asks for a sharp estimate in the case of the numbersRMg (M2n)and RMc (M2n). He means that these numbers are exponentially large in termsof the number of the vertices. This conjecture seems reasonable, because inthe case of abstract graphs the Ramsey-multiplicity tends to be large. Ac-cording to a conjecture of Harary and Prins, in the case of abstract graphsthe Ramsey-multiplicity can only be in some special cases non-exponential interms of the number of vertices (see [19]). That is why the following can seemunexpected:Theorem 10. RMg (M2n) = RMc (M2n) = 1.References[1] P. G. Agarwal and J. Pach, Combinatorial Geometry, Wiley, 1995.7[2] J. Borbély, On the higher dimensional generalization of a problem of Roth,Integers, Volume 13, A64, 2013.[3] J. Borbély, On the rainbow version of a problem of Roth, Journal of Com-binatorics and Number Theory 6 , Volume 1, 67-75, 2014.[4] J. Borbély, Some Ramsey-type asymptotic results concerning products, sub-mitted[5] J. Borbély, A rainbow-type asymptotic result concerning products, submit-ted[6] A. Cauchy, Recherches sur les nombres, Journal de l'École polytechnique,Volume 9, 99-116, 1813.[7] D. Conlon, Rainbow solutions of linear equations over Zp, Discrete Math-ematics , Volume 306, Issue 17, 2056-2063, 2006.[8] J. A. Dias da Silva and Y. O. Hamidoune, Cyclic spaces for Grassmanderivatives and additive theory, Bulletin of the London Mathematical So-ciety , Volume 26, 140-146, 1994.[9] P. Erd®s, Some unsolved problems, Publications of the Mathematical Insti-tute of the Hungarian Academy of Sciences, Volume A 6, 221-254, 1961.[10] P. Erd®s and A. Sárközy, On a conjecture of Roth and some related prob-lems II, in: Number Theory, Proceedings of The First Conference of TheCanadian Number Theory Association, edited by R. A. Mollin and Walterde Gruyter, Berlin-New York, 199-245, 1990.[11] P. Erd®s, A. Sárközy and Vera T. Sós, On a conjecture of Roth and somerelated problems I, Irregularities of partitions, Pap. Meet., Fert®d/Hung.,in: Algorithms and Combinatorics, Volume 8, 47-59, 1989.[12] L. Gerencsér, A. Gyárfás, On Ramsey-type problems, Annales Universi-tatis Scientarium Budapestiensis Lorand Eötvös, Sectio Mathematica X,167-170, 1967.8[13] R. L. Graham, B. L. Rotschild, J. H. Spencer, Ramsey Theory, Wiley,1980.[14] P. Gritzmann, B. Mohar and J. Pach and R. Pollack, Embedding a pla-nar triangulation with vertices at speciﬁed points, American MathematicalMonthly, Volume 98, 165-166, 1991.[15] D. S. Gunderson and V. Rödl, An alternate proof of Szemerédi's cubelemma using extremal hypergraphs, Diskrete Strukturen in der Mathematik,Preprintreihe, 1995.[16] D. S. Gunderson and V. Rödl, Extremal problems for aﬃne cubes of inte-gers, Combinatorics, Probability and Computing 7, Volume 1, 65-79, 1998.[17] R. K. Guy, Unsolved Problems In Number Theory, Springer-Verlag, 1980.[18] Y. Hamidoune and Ø. Rødseth, An inverse theorem mod p , Acta Arith-metica , Volume 92, 251-262, 2000.[19] F. Harary and G. Prins, Generalized Ramsey theory for graphs IV, Ramseymultiplicities, Networks, Volume 4 (1974), 163-173, 1974.[20] D. Hilbert, Über die Irreduzibilität ganzer rationaler Funktionen mit ganz-zähligen Koeﬃzienten, Crelle's Journal, Volume 110, 104-129, 1892.[21] V. Jungi¢ , J. Ne²et°il and R. Radoi£i¢, Rainbow Ramsey Theory, Integers,Volume 5, 2005.[22] G. Károlyi, Ramsey-type problems for geometric graphs, in: Thirty Essayson Geometric Graph Theory, Springer, 371-382, 2013.[23] G. Károlyi, J. Pach and G. Tóth, Ramsey-type results for geometric graphsI, Discrete and Computational Geometry, Volume 18, 247-255, 1997.[24] G. Károlyi, J. Pach, G. Tóth and P. Valtr, Ramsey-type results for geomet-ric graphs II, Discrete and Computational Geometry, Volume 20, 375-388,1998.9[25] J. B. Keller and C. Knessl, Partititon asymptotics from recursion equa-tions, SIAM Journal on Applied Mathematics, Volume 50, No. 2, 323-338,1990.[26] M. Kneser, Abschätzung der asymptotischen Dichte von Summenmengen,Mathematische Zeitschrift, Volume 58, 459-484, 1953.[27] B. M. Landman and A. Robertson, Ramsey Theory On The Integers,American Mathematical Society, 2004.[28] M. B. Nathanson, Additive Number Theory: The Classical Bases, Gradu-ate Texts in Mathematics, 1996.[29] M. B. Nathanson, Additive Number Theory: Inverse Problems and theGeometry of Sumsets, Graduate Texts in Mathematics, Volume 164, 1996.[30] M. B. Nathanson, Additive Number Theory: Inverse Problems and theGeometry of Sumsets, Graduate Texts in Mathematics, Volume 165, 1996.[31] J. Solymosi, Bounding multiplicative energy by the sumset, Advances inMathematics, Volume 222, Issue 2, 2009, 402-408. 1990.[32] E. Szemerédi, On sets of integers containing no k elements in arithmeticprogression, Acta Arithmetica 27, 199-245, 1975.[33] T. Tao and V. H. Vu, Additive Combinatorics, Cambridge Studies in Ad-vanced Mathematics, Volume 105, 2006.[34] A. G. Vosper, The critical pairs of subsets of a group of prime order,Journal of the London Mathematical Society, Volume 31, 200-205, 1956.[35] B. S. Webb (editor), Surveys in Combinatorics, London MathematicalSociety Lecture Note Series, Volume 327, 2005.10

Some Ramsey- and anti-Ramsey-type results in combinatorial number theory and geometry

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Some Ramsey- and anti-Ramsey-type results in combinatorial number theory and geometry

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