Ramsey-type Problem Research Articles

Curves in $\mathbb R^d$ intersecting every hyperplane at most $d+1$ times

By a curve in \mathbb R^d we mean a continuous map \gamma:I\to\mathbb R^d , where I\subset\mathbb R is a closed interval. We call a curve \gamma in \mathbb R^d\: (≤ k) -crossing if it intersects every hyperplane at most k times (counted with multiplicity). The (≤ d) -crossing curves in \mathbb R^d are often called convex curves and they form an important class; a primary example is the moment curve \{(t,t^2,\ldots,t^d):t\in[0,1]\} . They are also closely related to Chebyshev systems , which is a notion of considerable importance, e.g., in approximation theory. Our main result is that for every d there is M=M(d) such that every (≤ d+1) -crossing curve in \mathbb R^d can be subdivided into at most M\: (≤ d) -crossing curve segments. As a consequence, based on the work of Eliáš, Roldán, Safernová, and the second author, we obtain an essentially tight lower bound for a geometric Ramsey-type problem in \mathbb R^d concerning order-type homogeneous sequences of points, investigated in several previous papers.

Ramsey numbers for degree monotone paths

A path v1,v2,…,vm in a graph G is degree-monotone if deg(v1)≤deg(v2)≤⋯≤deg(vm) where deg(vi) is the degree of vi in G. Longest degree-monotone paths have been studied in several recent papers. Here we consider the Ramsey type problem for degree monotone paths. Denote by Mk(m) the minimum number M such that for all n≥M, in any k-edge coloring of Kn there is some 1≤j≤k such that the graph formed by the edges colored j has a degree-monotone path of order m. We prove several nontrivial upper and lower bounds for Mk(m).

Monochromatic Solutions for Multi-Term Unknowns

Given integers $$k\ge 2$$kź2, $$n \ge 2$$nź2, $$m \ge 2$$mź2 and $$ a_1,a_2,\ldots ,a_m \in {\mathbb {Z}}{\backslash }{\{0\}}$$a1,a2,ź,amźZ\{0}, and let $$f(z)= \sum _{j=0}^{n}c_jz^j$$f(z)=źj=0ncjzj be a polynomial of integer coefficients with $$c_n>0$$cn>0 and $$(\sum _{i=1}^ma_i)|f(z)$$(źi=1mai)|f(z) for some integer z. For a k-coloring of $$[N]=\{1,2,\ldots ,N\}$$[N]={1,2,ź,N}, we say that there is a monochromatic solution of the equation $$a_1x_1+a_2x_2+\cdots +a_mx_m=f(z)$$a1x1+a2x2+ź+amxm=f(z) if there exist pairwise distinct $$x_1,x_2,\ldots ,x_m\in [N]$$x1,x2,ź,xmź[N] all of the same color such that the equation holds for some $$z\in \mathbb {Z}$$zźZ. Problems of this type are often referred to as Ramsey-type problems. In this paper, it is shown that if $$a_i>0$$ai>0 for $$1\le i\le m$$1≤i≤m, then there exists an integer $$N_0=N(k,m,n)$$N0=N(k,m,n) such that for $$N\ge N_0$$NźN0, each k-coloring of [N] contains a monochromatic solution $$x_1,x_2,\ldots ,x_m$$x1,x2,ź,xm of the equation $$a_1x_1+a_2x_2+ \cdots +a_mx_m= f(z)$$a1x1+a2x2+ź+amxm=f(z). Moreover, if n is odd and there are $$a_i$$ai and $$a_j$$aj such that $$a_ia_j<0$$aiaj<0 for some $$1 \le i\ne j\le m$$1≤iźj≤m, then the assertion holds similarly.

On a Ramsey-type problem of Erdős and Pach

Erdős and Pach (1983) asked if there is some constant C>0 such that for any graph G on Ckln⁡k vertices either G or its complement G¯ has an induced subgraph on k vertices with minimum degree at least 12(k−1). They showed that the above statement holds with Ck2 in place of Ckln⁡k but that it does not hold with Ckln⁡k/ln⁡ln⁡k. We show that it holds with Ckln2⁡k, answering their question up to a ln⁡k factor.

Semi-algebraic Ramsey numbers

Given a finite point set P⊂Rd, a k-ary semi-algebraic relation E on P is a set of k-tuples of points in P determined by a finite number of polynomial equations and inequalities in kd real variables. The description complexity of such a relation is at most t if the number of polynomials and their degrees are all bounded by t. The Ramsey number Rkd,t(s,n) is the minimum N such that any N-element point set P in Rd equipped with a k-ary semi-algebraic relation E of complexity at most t contains s members such that every k-tuple induced by them is in E or n members such that every k-tuple induced by them is not in E.We give a new upper bound for Rkd,t(s,n) for k≥3 and s fixed. In particular, we show that for fixed integers d,t,sR3d,t(s,n)≤2no(1), establishing a subexponential upper bound on R3d,t(s,n). This improves the previous bound of 2nC1 due to Conlon, Fox, Pach, Sudakov, and Suk where C1 depends on d and t, and improves upon the trivial bound of 2nC2 which can be obtained by applying classical Ramsey numbers where C2 depends on s. As an application, we give new estimates for a recently studied Ramsey-type problem on hyperplane arrangements in Rd. We also study multi-color Ramsey numbers for triangles in our semi-algebraic setting, achieving some partial results.

Co-Nondeterminism in Compositions

The field of kernelization offers a rigorous way of studying the ubiquitous technique of data reduction and preprocessing for combinatorially hard problems. A widely accepted definition of useful data reduction is that of a polynomial kernelization where the output instance is guaranteed to be of size polynomial in some parameter of the input. The fairly recent development of a framework for kernelization lower bounds has made this notion even more attractive as we can now classify many problems into admitting or not admitting polynomial kernelizations. The central notion of the framework is that of a polynomial-time composition algorithm due to Bodlaender et al. (ICALP 2008, JSCC 2009): given t input instances, an or -composition algorithm returns a single-output instance with bounded parameter value that is yes if and only if one of t input instances is yes; it encodes the logical OR of the input instances. Based on a result of Fortnow and Santhanam (STOC 2008, JSCC 2011), Bodlaender et al. show that an or -composition for an NP-hard problem rules out polynomial kernelizations for it unless NP ⊆ coNP/poly (which is known to imply a collapse of the polynomial hierarchy). It is implicit in the work of Fortnow and Santhanam that even co-nondeterministic composition algorithms suffice to rule out polynomial kernelizations. This was first observed in unpublished work of Chen and Müller, and it is an explicit conclusion of recent results by Dell and van Melkebeek (STOC 2010). However, in contrast to the numerous applications of deterministic composition, the added power of co-nondeterminism has not yet been harnessed to obtain kernelization lower bounds. In this work, we present the first example of how co-nondeterminism can help to make a composition algorithm. We study the existence of polynomial kernels for a Ramsey-type problem where, given a graph G and an integer k , the question is whether G contains an independent set or a clique of size at least k . It was asked by Rod Downey whether this problem admits a polynomial kernelization with respect to k ; such a result would greatly speed up the computation of Ramsey numbers. We provide a co-nondeterministic composition based on embedding t instances into a single host graph H . The crux is that the host graph H needs to observe a bound of ℓ ∈ O (log t ) on both its maximum independent set and maximum clique size, while also having a cover of its vertex set by independent sets and cliques all of size ℓ; the co-nondeterministic composition is built around the search for such graphs. Thus, we show that, unless NP ⊆ coNP/poly, the problem does not admit a kernelization with polynomial size guarantee.

Ramsey-type results for semi-algebraic relations

A k-ary semi-algebraic relation E on R-d is a subset of R-kd, the set of k-tuples of points in R-d, which is determined by a finite number of polynomial inequalities in kd real variables. The description complexity of such a relation is at most t if d, k <= t and the number of polynomials and their degrees are all bounded by t. A set A subset of R-d is called homogeneous if all or none of the k-tuples from A satisfy E. A large number of geometric Ramseytype problems and results can be formulated as questions about finding large homogeneous subsets of sets in R-d equipped with semi-algebraic relations. In this paper, we study Ramsey numbers for k-ary semi-algebraic relations of bounded complexity and give matching upper and lower bounds, showing that they grow as a tower of height k-1. This improves upon a direct application of Ramsey's theorem by one exponential and extends a result of Alon, Pach, Pinchasi, Radoicic, and Sharir, who proved this for k = 2. We apply our results to obtain new estimates for some geometric Ramsey-type problems relating to order types and one-sided sets of hyperplanes. We also study the off-diagonal case, achieving some partial results.

Noncrossing Monochromatic Subtrees and Staircases in 0-1 Matrices

The following question is asked by the senior author (Gyárfás (2011)). What is the order of the largest monochromatic noncrossing subtree (caterpillar) that exists in every 2-coloring of the edges of a simple geometric Kn,n? We solve one particular problem asked by Gyárfás (2011): separate the Ramsey number of noncrossing trees from the Ramsey number of noncrossing double stars. We also reformulate the question as a Ramsey-type problem for 0-1 matrices and pose the following conjecture. Every n×n 0-1 matrix contains n−1 zeros or n−1 ones, forming a staircase: a sequence which goes right in rows and down in columns, possibly skipping elements, but not at turning points. We prove this conjecture in some special cases and put forward some related problems as well.

Diagonal forms of incidence matrices associated with [formula omitted]-uniform hypergraphs

We consider integer matrices Nt(h) whose rows are indexed by the t-subsets of an n-set and whose columns are all images of a particular column h under the symmetric group Sn. Earlier work has determined a diagonal form for Nt(h) when h has at least t ‘isolated vertices’ and the results were applied to the binary case of a zero-sum Ramsey-type problem of Alon and Caro involving t-uniform hypergraphs. This paper deals with the case that h does not have as many as t isolated vertices.

Turán and Ramsey Properties of Subcube Intersection Graphs

The discrete cube {0, 1}d is a fundamental combinatorial structure. A subcube of {0, 1}d is a subset of 2k of its points formed by fixing k coordinates and allowing the remaining d − k to vary freely. This paper is concerned with patterns of intersections among subcubes of the discrete cube. Two sample questions along these lines are as follows: given a family of subcubes in which no r + 1 of them have non-empty intersection, how many pairwise intersections can we have? How many subcubes can we have if among them there are no k which have non-empty intersection and no l which are pairwise disjoint?These questions are naturally expressed using intersection graphs. The intersection graph of a family of sets has one vertex for each set in the family with two vertices being adjacent if the corresponding subsets intersect. Let $\I(n,d)$ be the set of all n vertex graphs which can be represented as the intersection graphs of subcubes in {0, 1}d. With this notation our first question above asks for the largest number of edges in a Kr+1-free graph in $\I(n,d)$. As such it is a Turán-type problem. We answer this question asymptotically for some ranges of r and d. More precisely we show that if $(k+1)2^{\lfloor\frac{d}{k+1}\rfloor}&lt;n\leq k2^{\lfloor\frac{d}{k}\rfloor}$ for some integer k ≥ 2 then the maximum edge density is $\bigl(1-\frac{1}{k}-o(1)\bigr)$ provided that n is not too close to the lower limit of the range.The second question can be thought of as a Ramsey-type problem. The maximum such n can be defined in the same way as the usual Ramsey number but only considering graphs which are in $\I(n,d)$. We give bounds for this maximum n mainly concentrating on the case that l is fixed, and make some comparisons with the usual Ramsey number.

A Note on Universal and Canonically Coloured Sequences

A sequence X = {xi}ni=1 over an alphabet containing t symbols is t-universal if every permutation of those symbols is contained as a subsequence. Kleitman and Kwiatkowski showed that the minimum length of a t-universal sequence is (1 − o(1))t2. In this note we address a related Ramsey-type problem. We say that an r-colouring χ of the sequence X is canonical if χ(xi) = χ(xj) whenever xi = xj. We prove that for any fixedt the length of the shortest sequence over an alphabet of size t, which has the property that every r-colouring of its entries contains a t-universal and canonically coloured subsequence, is at most $cr^{\lfloor\frac{t}{2}\rfloor}$. This is best possible up to a multiplicative constant c independent of r.

Hypergraph Ramsey numbers

The Ramsey number r k ( s , n ) r_k(s,n) is the minimum N N such that every red-blue coloring of the k k -tuples of an N N -element set contains a red set of size s s or a blue set of size n n , where a set is called red (blue) if all k k -tuples from this set are red (blue). In this paper we obtain new estimates for several basic hypergraph Ramsey problems. We give a new upper bound for r k ( s , n ) r_k(s,n) for k ≥ 3 k \geq 3 and s s fixed. In particular, we show that \[ r 3 ( s , n ) ≤ 2 n s − 2 log ⁡ n , r_3(s,n) \leq 2^{n^{s-2}\log n}, \] which improves by a factor of n s − 2 / polylog n n^{s-2}/\textrm {polylog}\,n the exponent of the previous upper bound of Erdős and Rado from 1952. We also obtain a new lower bound for these numbers, showing that there is a constant c &gt; 0 c&gt;0 such that \[ r 3 ( s , n ) ≥ 2 c s n log ⁡ ( n s + 1 ) r_3(s,n) \geq 2^{c \, sn \, \log (\frac {n}{s}+1)} \] for all 4 ≤ s ≤ n 4 \leq s \leq n . For constant s s , this gives the first superexponential lower bound for r 3 ( s , n ) r_3(s,n) , answering an open question posed by Erdős and Hajnal in 1972. Next, we consider the 3 3 -color Ramsey number r 3 ( n , n , n ) r_3(n,n,n) , which is the minimum N N such that every 3 3 -coloring of the triples of an N N -element set contains a monochromatic set of size n n . Improving another old result of Erdős and Hajnal, we show that \[ r 3 ( n , n , n ) ≥ 2 n c log ⁡ n . r_3(n,n,n) \geq 2^{n^{c \log n}}. \] Finally, we make some progress on related hypergraph Ramsey-type problems.

Using the incompressibility method to obtain local lemma results for Ramsey-type problems

We reveal a connection between the incompressibility method and the Lovász local lemma in the context of Ramsey theory. We obtain bounds by repeatedly encoding objects of interest and thereby compressing strings. The method is demonstrated on the example of van der Waerden numbers. In particular we reprove that w ( k ; c ) ⩾ c k − 3 k ⋅ k − 1 k . The method is applicable to obtain lower bounds of Ramsey numbers, large transitive subtournaments and other Ramsey phenomena as well.

Monochromatic and Heterochromatic Subgraphs in Edge-Colored Graphs - A Survey

Nowadays the term monochromatic and heterochromatic (or rainbow, multicolored) subgraphs of an edge-colored graph appeared frequently in literature, and many results on this topic have been obtained. In this paper, we survey results on this subject. We classify the results into the following categories: vertex-partitions by monochromatic subgraphs, such as cycles, paths, trees; vertex partition by some kinds of heterochromatic subgraphs; the computational complexity of these partition problems; some kinds of large monochromatic and heterochromatic subgraphs. We have to point out that there are a lot of results on Ramsey type problem of monochromatic and heterochromatic subgraphs. However, it is not our purpose to include them in this survey because this is slightly different from our topics and also contains too large amount of results to deal with together. There are also some interesting results on vertex-colored graphs, but we do not include them, either.

On Ramsey Type Problems in Combinatorial Geometry

In this paper, we will give a short survey of various old and new results on Ramsey type problems in combinatorial geometry.

On a series of Ramsey-type problems in combinatorial geometry

On a series of Ramsey-type problems in combinatorial geometry

Connecting colored point sets

We study the following Ramsey-type problem. Let S = B ∪ R be a two-colored set of n points in the plane. We show how to construct, in O ( n log n ) time, a crossing-free spanning tree T ( B ) for B, and a crossing-free spanning tree T ( R ) for R, such that both the number of crossings between T ( B ) and T ( R ) and the diameters of T ( B ) and T ( R ) are kept small. The algorithm is conceptually simple and is implementable without using any non-trivial data structure. This improves over a previous method in Tokunaga [Intersection number of two connected geometric graphs, Inform. Process. Lett. 59 (1996) 331–333] that is less efficient in implementation and does not guarantee a diameter bound. Implicit to our approach is a new proof for the result in the reference above on the minimum number of crossings between T ( B ) and T ( R ) .

Properly colored subgraphs and rainbow subgraphs in edge‐colorings with local constraints

AbstractWe consider a canonical Ramsey type problem. An edge‐coloring of a graph is called m‐good if each color appears at most m times at each vertex. Fixing a graph G and a positive integer m, let f(m, G) denote the smallest n such that every m‐good edge‐coloring of Kn yields a properly edge‐colored copy of G, and let g(m, G) denote the smallest n such that every m‐good edge‐coloring of Kn yields a rainbow copy of G. We give bounds on f(m, G) and g(m, G). For complete graphs G = Kt, we have c1mt2/ln t ≤ f(m, Kt) ≤ c2mt2, and cmt3/ln t ≤ g(m, Kt) ≤ cmt3/ln t, where c1, c2, c, c are absolute constants. We also give bounds on f(m, G) and g(m, G) for general graphs G in terms of degrees in G. In particular, we show that for fixed m and d, and all sufficiently large n compared to m and d, f(m, G) = n for all graphs G with n vertices and maximum degree at most d. © 2003 Wiley Periodicals, Inc. Random Struct. Alg., 2003

A lower bound for certain Ramsey type problems

One of the main results in this article is to present a new and simple proof of the following theorem; The Diophantine equation a 1/y 1 + a 2/y 2 + ··· + an /y n = 0, where yi ≠ 0 are n unknown variables and ai 's are non-zero integers, is regular if and only if there exists a non empty subset I of {1, 2, ···, n} such that ∑ i∈I ai = 0. In the rest of this article, we study a particular case when n = 3 in the above equation. Also, we define f(r) to be the largest integer such that if the integers in [1,f(r)] are r coloured, then no monochromatic solution to the Diophantine equation 1/x + 1/y = 1/z, x, y, z can exist. Then we prove that f(r) ≥ 2 r-3.192 - 1 and f(r + 2) ≥ 4 (f(r) + √(f(r)/5).

Optimal risk allocation for regulated monopolies and consumers

The model shows how a regulated monopolist’s price should change as random cost and demand parameters are revealed. The regulator has a Ramsey-type problem. With a linear tariff a trade-off between allocative efficiency and risk sharing typically exists. The attitudes of the consumer and the firm to both income and price risk determine how the price should move. Sufficient conditions are found for price adjustment schemes used in practice to be optimal. These schemes include full, partial and zero pass-through of marginal costs, and price caps, average cost pricing and caps on total revenue for demand risk.