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Mathematics and statistics journals publish papers on the theory and application of mathematics, statistics, and probability. Most mathematics journals have a broad scope that encompasses most mathematical fields. These commonly include logic and foundations, algebra and number theory, analysis (including differential equations, functional analysis and operator theory), geometry, topology, combinatorics, probability and statistics, numerical analysis and computation theory, mathematical physics, etc.
Mathematics and Statistics
We prove that, when 𝑛 goes to infinity, Kostant’s problem has negative answer for almost all simple highest weight modules in the principal block of the BGG category O for the Lie algebra sl𝑛(ℂ).
In this article, we introduce a non-negative integer-valued function that measures the obstruction for converting topological isotopy between two Legendrian knots into a Legendrian isotopy. We refer to this function as the Cost function. We show that the Cost function induces a metric on the set of topologically isotopic Legendrian knots. Hence, the set of topologically isotopic Legendrian knots can be seen as a graph with path-metric given by the Cost function. Legendrian simple knot types are shown to be characterized using the Cost function. We also get a quantitative version of Fuchs–Tabachnikov’s Theorem that says any two Legendrian knots in (𝕊3, 𝜉𝑠𝑡𝑑) in the same topological knot type become Legendrian isotopic after sufficiently many stabilizations [8]. We compute the Cost function for Legendrian simple knots (for example torus knots) and we note the behavior of Cost function for twist knots and cables of torus knots (some of which are Legendrian non-simple). We also construct examples of Legendrian representatives of 2-bridge knots and compute the Cost between them. Further, we investigate the behavior of the Cost function under the connect sum operation. We conclude with some questions about the Cost function, its relation with the standard contact structure, and the topological knot type.
Recent results have provided important functional generalizations, extensions and improvements of the Hardy and Levinson integral inequalities. However, they require some assumptions on the main functions, such as monotonicity or convexity assumptions, which remain somewhat restrictive. In this article, we propose two new ideas of functional generalizations, one based on a series expansion approach and the other on an integral approach. Both achieve the goal of offering adaptable generalizations and extensions of the Hardy and Levinson integral inequalities. They are formulated in two different general theorems, which are proved in detail. Several examples of new integral inequalities are derived.
Breuer and Klivans defined a diverse class of scheduling problems in terms of Boolean formulas with atomic clauses that are inequalities. We consider what we call graph-like scheduling problems. These are Boolean formulas that are conjunctions of disjunctions of atomic clauses (𝑥𝑖 ≠ 𝑥𝑗). These problems generalize proper coloring in graphs and hypergraphs. We focus on the existence of a solution with all 𝑥 i taking the value of 0 or 1 (i.e. problems analogous to the bipartite case). When a graph-like scheduling problem has such a solution, we say it has property B just as is done for 2-colorable hypergraphs. We define the notion of a 𝜆-uniform graph-like scheduling problem for any integer partition 𝜆. Some bounds are attained for the size of the smallest 𝜆-uniform graph-like scheduling problems without B. We make use of both random and constructive methods to obtain bounds. Just as in the case of hypergraphs finding tight bounds remains an open problem.
Let {𝐿𝑛}≥0 be the sequence of Lucas numbers. In this paper, we determine all Lucas numbers that are palindromic concatenations of two distinct repdigits.
We study the “no-dimensional” analogue of Helly’s theorem in Banach spaces. Specifically, we obtain the following no-dimensional Helly-type results for uniformly convex Banach spaces: Helly’s theorem, fractional Helly’s theorem, colorful Helly’s theorem, and colorful fractional Helly’s theorem.
The combinatorial part of the proofs for these Helly-type results is identical to the Euclidean case as presented in [2]. The primary difference lies in the use of a certain geometric inequality in place of the Pythagorean theorem. This inequality can be explicitly expressed in terms of the modulus of convexity of a Banach space.
We revisit the algorithmic problem of finding a triangle in a graph (Triangle Detection), and examine its relation to other problems such as 3Sum, Independent Set, and Graph Coloring. We obtain several new algorithms:
(I) A simple randomized algorithm for finding a triangle in a graph. As an application, we study a question of Pˇatraşcu (2010) regarding the triangle detection problem.
(II) An algorithm which given a graph 𝐺 = (𝑉 , 𝐸) performs one of the following tasks in 𝑂(𝑚 + 𝑛) (i.e., linear) time: (i) compute a Ω(1/√𝑛)-approximation of a maximum independent set in 𝐺 or (ii) find a triangle in 𝐺. The run-time is faster than that for any previous method for each of these tasks.
(III) An algorithm which given a graph 𝐺 = (𝑉 , 𝐸) performs one of the following tasks in 𝑂(𝑚+𝑛3/2) time: (i) compute √𝑛-approximation for Graph Coloring of 𝐺 or (ii) find a triangle in 𝐺. The run-time is faster than that for any previous method for each of these tasks on dense graphs, with 𝑚 = (𝑛9/8).
(IV) Results (II) and (III) above suggest the following broader research direction: if it is difficult to find (A) or (B) separately, can one find one of the two efficiently? This motivates the dual pair concept we introduce. We provide several instances of dual-pair approximation, relating Longest Path, (1,2)-TSP, and other NP-hard problems.
A question of Erdős asked whether there exists a set of 𝑛 points such that 𝑐 ⋅ 𝑛 distances occur more than 𝑛 times. We provide an affirmative answer to this question, showing that there exists a set of 𝑛 points such that
The Erdős Matching Conjecture states that the maximum size 𝑓 (𝑛, 𝑘, 𝑠) of a family
In this short note, we are concerned with the case 𝑠 ≥ 3 fixed, 𝑘 tending to infinity and 𝑛 ∈ {𝑠𝑘, 𝑠𝑘 + 1}. For 𝑛 = 𝑠𝑘, we show the stability of the unique extremal construction of size
A long standing Total Coloring Conjecture (TCC) asserts that every graph is total colorable using its maximum degree plus two colors. A graph is type-1 (or type-2) if it has a total coloring using maximum degree plus one (or maximum degree plus two) colors. For a graph 𝐺 with 𝑚 vertices and for a family of graphs {𝐻1, 𝐻2, … , 𝐻𝑚}, denote
We study the property of Kelley and the property of Kelley weakly on Hausdorff continua. We extend results known for metric continua to the class of Hausdorff continua. We also present new results about these properties.
The aim of this paper is to study the interrelationship between various forms of (F, G)-shadowing property and represent it through the diagram. We show that asymptotic shadowing is equivalent to (ℕ0, F 𝑐𝑓 )-shadowing property and that (ℕ0, F 𝑐𝑓 )-shadowing implies (F 𝑐𝑓 , F 𝑐𝑓 )-shadowing. Necessary examples are discussed to support the diagram. We also give characterization for maps to have the (F, G)-shadowing property through the shift map on the inverse limit space. Further, we relate the (F, G)-shadowing property to the positively F 𝑠-expansive map. Also, we obtain the necessary and sufficient condition for the identity map to have (ℕ0, F 𝑡)-shadowing property.
We prove that the filtered GRID invariants of Legendrian links in link Floer homology, and consequently their associated invariants in the spectral sequence, obstruct decomposable Lagrangian cobordisms in the symplectization of the standard contact structure on ℝ3, strengthening a result by Baldwin, Lidman, and the fifth author.
In [3] it is shown, answering a question of Jordán and Nguyen [9], that universal rigidity of a generic bar-joint framework in ℝ1 depends on more than the ordering of the vertices. The graph 𝐺 that was used in that paper is a ladder with three rungs. Here we provide a general answer when that ladder with three rungs in the line is universally rigid and when it is not.
In this paper the author studies the problem of finding the farthest points in an intersection of balls to a given point 𝐶0. A polynomial algorithm is presented which solves the problem under the conditions that the given point is outside of the convex hull of the balls centers. It is shown that in this particular case the problem of finding the smallest ball centered in 𝐶0 which includes the intersection of balls is actually convex.
In this article, we present new results on specific cases of a general Young integral inequality established by Páles in 1990. Our initial focus is on a bivariate function, defined as the product of two univariate and separable functions. Based on this, some new results are established, including particular Young integral-type inequalities and some upper bounds on the corresponding absolute errors. The precise role of the functions involved in this context is investigated. Several applications are presented, including one in the field of probability theory. We also introduce and study reverse variants of our inequalities. Another important contribution is to link the setting of the general Young integral inequality established by Páles to a probabilistic framework called copula theory. We show that this theory provides a wide range of functions, often dependent on adjustable parameters, that can be effectively applied to this inequality. Some illustrative graphics are provided. Overall, this article broadens the scope of bivariate inequalities and can serve related purposes in analysis, probability and statistics, among others.
Let 𝑛, 𝑠, 𝑣 be positive integers and F ⊂ 2[𝑛]. Suppose that the union of any 𝑠 sets of F has size at most 𝑠𝑣 and 𝑛 ≥ 2𝑠+3𝑣. The main result implies the best possible bound
In this work we single out a scheduling problem in which tasks are coupled and the time delay between the first and second members of the couple is fixed by technological constraints. We will show that this scheduling problem can be reduced to the question to decide if a tactically constructed 𝑘-partite auxiliary graph contains a 𝑘-clique. We will point out that before submitting the auxiliary graph to a clique solver it is expedient to carry out various inspections in order to delete nodes and edges of the graph and consequently speed up the computations. In the lack of theoretical tools we will carry out numerical experiments to test the practicality of the clique approach.
In this paper, we define the discretized Voros–Li coefficients associated to the zeta function on function fields of genus 𝑔 over a finite fields 𝔽𝑞. Furthermore, we give a finite sum representation, an integral formula and an asymptotic formula for these coefficients.
In the case of symmetries with respect to 𝑛 independent linear hyperplanes, a stability versions of the Logarithmic Brunn–Minkowski Inequality and the Logarithmic Minkowski Inequality for convex bodies are established.
In this article, we use the idea of “negation” to construct new unit distributions, i.e., continuous distributions with support equal to the unit interval [0, 1]. A notable feature of these distributions is that they have opposite shape properties to the unit distributions from which they are derived; “opposite” in the sense that, from a graphical point of view, a certain horizontal symmetry is operated. We then examine the main properties of these negation-type distributions, including distributional functions, moments, and entropy measures. Finally, concrete examples are described, namely the negation-type power distribution, the negation-type [0, 1]-truncated exponential distribution, the negation-type truncated [0, 1]-sine distribution, the negation-type [0, 1]-truncated Lomax distribution, the negation-type Kumaraswamy distribution, and the negation-type beta distribution. Some of their properties are studied, also with the help of graphics that highlight their original modeling behavior. After the analysis, it appears that the negation-type Kumaraswamy distribution stands out from the others by combining simplicity with a high degree of flexibility, in a sense completing the famous Kumaraswamy distribution. Overall, our results enrich the panel of unit distributions available in the literature with an innovative approach.
High proved the following theorem. If the intersections of any two congruent copies of a plane convex body are centrally symmetric, then this body is a circle. In our paper we extend the theorem of High to the sphere and the hyperbolic plane. Let us have in 𝑆2, ℝ2 or 𝐻2 a pair of convex bodies (for 𝑆2 different from 𝑆2), such that the intersections of any congruent copies of them are centrally symmetric. Then our bodies are congruent circles. If the intersections of any congruent copies of them are axially symmetric, then our bodies are (incongruent) circles. Let us have in 𝑆2, ℝ2 or 𝐻2 proper closed convex subsets 𝐾, 𝐿 with interior points, such that the numbers of the connected components of the boundaries of 𝐾 and 𝐿 are finite. If the intersections of any congruent copies of 𝐾 and 𝐿 are centrally symmetric, then 𝐾 and 𝐿 are congruent circles, or, for ℝ2, parallel strips. For ℝ2 we exactly describe all pairs of such subsets 𝐾, 𝐿, whose any congruent copies have an intersection with axial symmetry (there are five cases).
This paper is mainly about direct summand right ideals of nearrings with 𝐷𝐶𝐶𝑁 which cannot be expressed as a non-trivial direct sum. A fairly natural condition (Φ-irreducibility) makes it possible to study these right ideals in reasonable depth. It turns out they are either very ring like or right ideals (called shares) controlling considerable nearring structure. The two cases are studied in some detail. A surprising feature of the last section is that, with weak hypercentrality present, the nearring is a unique finite direct sum of these right ideals if, and only if, all such right ideals are ideals.
In the present paper we aim to calculate with the exclusive use of real methods, an atypical harmonic series with a weight 4 structure, featuring the harmonic number of the kind 𝐻2𝑘. Very simple relations and neat results are considered for the evaluation of the main series.
This article describes a general analytical derivation of the Fuss’ relation for bicentric polygons with an odd number of vertices. In particular, we derive the Fuss’ relations for the bicentric tridecagon and the bicentric pentadecagon.
We improve the lower bound on the translative covering density of tetrahedra found by Y. Li, M. Fu and Y. Zhang. Our method improves the bound from 1.00122 to 1.0075, but also shows the existence of similar lower density bounds for any polyhedron which has a face without opposite parallel face or edge.
In this paper, we consider the asymptotic behaviour of the expectation of the number of vertices of a uniform random spherical disc-polygon. This provides a connection between the corresponding results in spherical convexity, and in Euclidean spindle-convexity, where the expectation tends to the same constant. We also extend the result to a more general case, where the random points generating the uniform random disc-polygon are chosen from spherical convex disc with smooth boundary.
We prove existence of Helly numbers for crystals and for cut-and-project sets with convex windows. Also we show that for a two-dimensional crystal consisting of 𝑘 copies of a single lattice the Helly number does not exceed 𝑘 + 6.
Consider an arrangement of 𝑛 congruent zones on the 𝑑-dimensional unit sphere 𝑆𝑑−1, where a zone is the intersection of an origin symmetric Euclidean plank with 𝑆𝑑−1. We prove that, for sufficiently large 𝑛, it is possible to arrange 𝑛 congruent zones of suitable width on 𝑆𝑑−1 such that no point belongs to more than a constant number of zones, where the constant depends only on the dimension and the width of the zones. Furthermore, we also show that it is possible to cover 𝑆𝑑−1 by 𝑛 congruent zones such that each point of 𝑆𝑑−1 belongs to at most 𝐴𝑑 ln 𝑛 zones, where the 𝐴𝑑 is a constant that depends only on 𝑑. This extends the corresponding 3-dimensional result of Frankl, Nagy and Naszódi [8]. Moreover, we also examine coverings of 𝑆𝑑−1 with congruent zones under the condition that each point of the sphere belongs to the interior of at most 𝑑 − 1 zones.
In this note we introduce a pseudometric on closed convex planar curves based on distances between normal lines and show its basic properties. Then we use this pseudometric to give a shorter proof of the theorem by Pinchasi that the sum of perimeters of 𝑘 convex planar bodies with disjoint interiors contained in a convex body of perimeter 𝑝 and diameter 𝑑 is not greater than 𝑝 + 2(𝑘 − 1)𝑑.
We study translative arrangements of centrally symmetric convex domains in the plane (resp., of congruent balls in the Euclidean 3-space) that neither pack nor cover. We define their soft density depending on a soft parameter and prove that the largest soft density for soft translative packings of a centrally symmetric convex domain with 3-fold rotational symmetry and given soft parameter is obtained for a proper soft lattice packing. Furthermore, we show that among the soft lattice packings of congruent soft balls with given soft parameter the soft density is locally maximal for the corresponding face centered cubic (FCC) lattice.
Let 𝑀𝑘 be the 𝑘-th Mulatu number. Let 𝑟, 𝑠 be non-zero integers with 𝑟 ≥ 1 and 𝑠 ∈ {−1, 1}, let {𝑈𝑛}𝑛≥0 be the generalized Lucas sequence and {𝑉𝑛}𝑛≥0 its companion given respectively by 𝑈𝑛+2 = 𝑟𝑈𝑛+1 + 𝑠𝑈𝑛 and 𝑉𝑛+2 = 𝑟𝑉𝑛+1 + 𝑠𝑉𝑛, with 𝑈0 = 0, 𝑈1 = 1, 𝑉0 = 2, 𝑉1 = 𝑟. In this paper, we give effective bounds for the solutions of the following Diophantine equations 𝑀𝑘 = 𝑈𝓁𝑈𝑚𝑈𝑛 and 𝑀𝑘 = 𝑉𝓁𝑉𝑚𝑉𝑛, where 𝓁, 𝑚, 𝑛 and 𝑘 are nonnegative integers and 𝓁 ≤ 𝑚 ≤ 𝑛. Then, we explicitly solve the above Diophantine equations for the Fibonacci, Pell, Balancing sequences and their companions respectively.
Let 𝑛 ≥ 2. A continuous 𝑛-linear form 𝑇 on a Banach space 𝐸 is called norm-peak if there is a unique (𝑥1, … , 𝑥𝑛) ∈ 𝐸𝑛 such that ║𝑥1║ = … = ║𝑥𝑛║ = 1 and for the multilinear operator norm it holds ‖𝑇 ‖ = |𝑇 (𝑥1, … , 𝑥𝑛)|.
Let 0 ≤ 𝜃 ≤
In this note, we characterize all norm-peak multilinear forms on
In this paper we introduce a construction for a weighted CW complex (and the associated lattice cohomology) corresponding to partially ordered sets with some additional structure. This is a generalization of the construction seen in [4] where we started from a system of subspaces of a given vector space. We then proceed to prove some basic properties of this construction that are in many ways analogous to those seen in the case of subspaces, but some aspects of the construction result in complexities not present in that scenario.
Let F be a nonempty family of graphs. A graph 𝐺 is called F -free if it contains no graph from F as a subgraph. For a positive integer 𝑛, the planar Turán number of F, denoted by exp (𝑛, F), is the maximum number of edges in an 𝑛-vertex F -free planar graph.
Let Θ𝑘 be the family of Theta graphs on 𝑘 ≥ 4 vertices, that is, graphs obtained by joining a pair of non-consecutive of a 𝑘-cycle with an edge. Lan, Shi and Song determined an upper bound exp (𝑛, Θ6) ≤ 18𝑛/7−36𝑛/7, but for large 𝑛, they did not verify that the bound is sharp. In this paper, we improve their bound by proving exp (𝑛, Θ6) ≤ 18𝑛/−48𝑛/7 and then we demonstrate the existence of infinitely many positive integer 𝑛 and an 𝑛-vertex Θ6-free planar graph attaining the bound.
Suppose that 𝑇 (𝛼, 𝛽) is an obtuse triangle with base length 1 and with base angles 𝛼 and 𝛽 (where 𝛽 > 90◦). In this note a tight lower bound of the sum of the areas of squares that can parallel cover 𝑇 (𝛼, 𝛽) is given. This result complements the previous lower bound obtained for the triangles with the interior angles at the base of the measure not greater than 90◦.
We show that every positroid of rank 𝑟 ≥ 2 has a good coline. Using the definition of the chromatic number of oriented matroid introduced by J. Nešetřil, R. Nickel, and W. Hochstättler, this shows that every orientation of a positroid of rank at least 2 is 3-colorable.
Let 𝑇 be a tree, a vertex of degree one is called a leaf. The set of all leaves of 𝑇 is denoted by Leaf(𝑇). The subtree 𝑇 − Leaf(𝑇) of 𝑇 is called the stem of 𝑇 and denoted by Stem(𝑇). A tree 𝑇 is called a caterpillar if Stem(𝑇) is a path. In this paper, we give two sufficient conditions for a connected graph to have a spanning tree whose stem is a caterpillar. We also give some examples to show that these conditions are sharp.
We revisit the problem of property testing for convex position for point sets in ℝ𝑑. Our results draw from previous ideas of Czumaj, Sohler, and Ziegler (2000). First, their testing algorithm is redesigned and its analysis is revised for correctness. Second, its functionality is expanded by (i) exhibiting both negative and positive certificates along with the convexity determination, and (ii) significantly extending the input range for moderate and higher dimensions.
The behavior of the randomized tester on input set 𝑃 ⊂ ℝ𝑑 is as follows: (i) if 𝑃 is in convex position, it accepts; (ii) if 𝑃 is far from convex position, with probability at least 2/3, it rejects and outputs a (𝑑 +2)-point witness of non-convexity as a negative certificate; (iii) if 𝑃 is close to convex position, with probability at least 2/3, it accepts and outputs a subset in convex position that is a suitable approximation of the largest subset in convex position. The algorithm examines a sublinear number of points and runs in subquadratic time for every fixed dimension 𝑑.
We prove zero density theorems for Dedekind zeta functions in the vicinity of the line Re s = 1, improving an earlier result of W. Staś.
A positive integer
Let (𝑃𝑛)𝑛≥0 and (𝑄𝑛)𝑛≥0 be the Pell and Pell–Lucas sequences. Let 𝑏 be a positive integer such that 𝑏 ≥ 2. In this paper, we prove that the following two Diophantine equations 𝑃𝑛 = 𝑏𝑑𝑃𝑚 + 𝑄𝑘 and 𝑃𝑛 = 𝑏𝑑𝑄𝑚 + 𝑃𝑘 with 𝑑, the number of digits of 𝑃𝑘 or 𝑄𝑘 in base 𝑏, have only finitely many solutions in nonnegative integers (𝑚, 𝑛, 𝑘, 𝑏, 𝑑). Also, we explicitly determine these solutions in cases 2 ≤ 𝑏 ≤ 10.
Grätzer and Lakser asked in the 1971 Transactions of the American Mathematical Society if the pseudocomplemented distributive lattices in the amalgamation class of the subvariety generated by 𝟐𝑛 ⊕ 𝟏 can be characterized by the property of not having a *-homomorphism onto 𝟐𝑖 ⊕ 𝟏 for 1 < 𝑖 < 𝑛.
In this article, their question from 1971 is answered.
This paper introduces and examines the concept of a *-Rickart *-ring, and proves that every Rickart *-ring is also a *-Rickart *-ring. A necessary and sufficient condition for a *-Rickart *-ring to be a Rickart *-ring is also provided. The relationship between *-Rickart *-rings and *-Baer *-rings is investigated, and several properties of *-Rickart *-rings are presented. The paper demonstrates that the property of *-Rickart extends to both the center and *-corners of a *-ring, and investigates the extension of a *-Rickart *-ring to its polynomial *-ring. Additionally, *-Rickart *-rings with descending chain condition on *-biideals are studied, and all *-Rickart (*-Baer) *-rings with finitely many elements are classified.
Very recently, the authors in [5] proposed the exponential-type operator connected with
A bi-cyclic 4-polytope in ℝ4 was introduced by Z. Smilansky as the convex hull of evenly spaced points on a generalized trigonometric moment curve in ℝ4. We present combinatorial geometric conditions that yield the face lattices of a class of such 4-polytopes.
A classical result of Dowker (Bull. Amer. Math. Soc. 50: 120-122, 1944) states that for any plane convex body 𝐾, the areas of the maximum (resp. minimum) area convex 𝑛-gons inscribed (resp. circumscribed) in 𝐾 is a concave (resp. convex) sequence. It is known that this theorem remains true if we replace area by perimeter, or convex 𝑛-gons by disk-𝑛-gons, obtained as the intersection of 𝑛 closed Euclidean unit disks. It has been proved recently that if 𝐶 is the unit disk of a normed plane, then the same properties hold for the area of 𝐶-𝑛-gons circumscribed about a 𝐶-convex disk 𝐾 and for the perimeters of 𝐶-𝑛-gons inscribed or circumscribed about a 𝐶-convex disk 𝐾, but for a typical origin-symmetric convex disk 𝐶 with respect to Hausdorff distance, there is a 𝐶-convex disk 𝐾 such that the sequence of the areas of the maximum area 𝐶-𝑛-gons inscribed in 𝐾 is not concave. The aim of this paper is to investigate this question if we replace the topology induced by Hausdorff distance with a topology induced by the surface area measure of the boundary of 𝐶.
A 𝑞-graph with 𝑒 edges and 𝑛 vertices is defined as an 𝑒 × 𝑛 matrix with entries from {0, … , 𝑞}, such that each row of the matrix (called a 𝑞-edge) contains exactly two nonzero entries. If 𝐻 is a 𝑞-graph, then 𝐻 is said to contain an 𝑠-copy of the ordinary graph 𝐹, if a set 𝑆 of 𝑞-edges can be selected from 𝐻 such that their intersection graph is isomorphic to 𝐹, and for any vertex 𝑣 of 𝑆 and any two incident edges 𝑒, 𝑓 ∈ 𝑆 the sum of the entries of 𝑒 and 𝑓 is at least 𝑠. The extremal number ex(𝑛, 𝐹, 𝑞, 𝑠) is defined as the maximal number of edges in an 𝑛-vertex 𝑞-graph such that it does not contain contain an 𝑠-copy of the forbidden graph 𝐹.
In the present paper, we reduce the problem of finding ex(𝑛, 𝐹, 𝑞, 𝑞 + 1) for even 𝑞 to the case 𝑞 = 2, and determine the asymptotics of ex(𝑛, 𝐶2𝑘+1, 𝑞, 𝑞 + 1).
For positive integers 𝑛, 𝑟, 𝑠 with 𝑟 > 𝑠, the set-coloring Ramsey number 𝑅(𝑛; 𝑟, 𝑠) is the minimum 𝑁 such that if every edge of the complete graph 𝐾𝑁 receives a set of 𝑠 colors from a palette of 𝑟 colors, then there is guaranteed to be a monochromatic clique on 𝑛 vertices, that is, a subset of 𝑛 vertices where all of the edges between them receive a common color. In particular, the case 𝑠 = 1 corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on 𝑅(𝑛; 𝑟, 𝑠) which imply that 𝑅(𝑛; 𝑟, 𝑠) = 2Θ(𝑛𝑟) if 𝑠/𝑟 is bounded away from 0 and 1. The upper bound extends an old result of Erdős and Szemerédi, who treated the case 𝑠 = 𝑟 − 1, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs.
