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Mathematics and Statistics

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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).

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Studia Scientiarum Mathematicarum Hungarica
Authors:
David Conlon
,
Jacob Fox
,
Xiaoyu He
,
Dhruv Mubayi
,
Andrew Suk
, and
Jacques Verstraëte

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.

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We consider a function from the Euclidean three space whose zero set is the image of the standard cuspidal edge. The composition of a parametrized singular surface in the three space with this function provides an approximation of the surface by the standard cuspidal edge. Taking a look at singularities of this composition, we study various approximations of singular surfaces like the cross cap, the generalized cuspidal edge and the swallowtail by standard cuspidal edges.

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Based on Peter’s work from 2003, quadrilaterals can be characterized in the following way: “among all quadrilaterals with given side lengths 𝑎, 𝑏, 𝑐 and 𝑑, those of the largest possible area are exactly the cyclic ones”. In this paper, we will give the corresponding characterization for every polygon, by means of quasicyclic polygons properties.

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In this paper, some basic characterizations of a weighted Bloch space with the differentiable strictly positive weight 𝜔 on the unit disc are given, including the growth, the higher order or free derivative descriptions, and integral characterizations of functions in the space.

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We present examples of multiplicative semigroups of positive reals (Beurling’s generalized integers) with gaps bounded from below.

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In this paper, we propose some new positive linear approximation operators, which are obtained from a composition of certain integral type operators with certain discrete operators. It turns out that the new operators can be expressed in discrete form. We provide representations for their coefficients. Furthermore, we study their approximation properties and determine their moment generating functions, which may be useful in finding several other convergence results in different settings.

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Let 𝑓 be a normalized primitive cusp form of even integral weight for the full modular group Γ = 𝑆𝐿(2, ℤ). In this paper, we investigate upper bounds for the error terms related to the average behavior of Fourier coefficients 𝜆𝑓 ⊗𝑓 ⊗⋯⊗𝑙𝑓 (𝑛) of 𝑙-fold product 𝐿-functions, where 𝑓 ⊗ 𝑓 ⊗ ⋯ ⊗𝑙 𝑓 denotes the 𝑙-fold product of 𝑓. These results improves and generalizes the recent developments of Venkatasubbareddy and Sankaranarayanan [41]. We also provide some other similar results related to the error terms of general product 𝐿-functions.

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The paper is devoted to some extremal problems for convex polygons on the Euclidean plane, related to the concept of self Chebyshev radius for the polygon boundary. We consider a general problem of minimization of the perimeter among all 𝑛-gons with a fixed self Chebyshev radius of the boundary. The main result of the paper is the complete solution of the mentioned problem for 𝑛 = 4: We proved that the quadrilateral of minimum perimeter is a so called magic kite, that verified the corresponding conjecture by Rolf Walter.

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Let 𝑛 ≥ 2 be an integer. The graph G n ¯ is obtained by letting all the elements of {0, … , 𝑛 − 1} to be the vertices and defining distinct vertices 𝑥 and 𝑦 to be adjacent if and only if gcd(𝑥 + 𝑦, 𝑛) ≠ 1. In this paper, we give some bounds for the Castelnuovo–Mumford regularity of the edge ideals and their powers for G n ¯ .

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