An explicit construction for a generalized Ramsey problem
نویسنده
چکیده
An explicit coloring of the edges of Kn is constructed such that every copy of K4 has at least four colors on its edges. As n →∞, the number of colors used is n. This improves upon the previous bound of O(n) due to Erdős and Gyárfás obtained by probabilistic methods. The exponent 1/2 is optimal, since it is known that at least Ω(n) colors are required in such a coloring. The coloring is related to constructions giving lower bounds for the multicolor Ramsey number rk(C4). It is more complicated however, because of restrictions imposed on interactions between color classes.
منابع مشابه
A NEW APPROACH TO THE SOLUTION OF SENSITIVITY MINIMIZATION IN LINEAR STATE FEEDBACK CONTROL
In this paper, it is shown that by exploiting the explicit parametric state feedback solution, it is feasible to obtain the ultimate solution to minimum sensitivity problem. A numerical algorithm for construction of a robust state feedback in eigenvalue assignment problem for a controllable linear system is presented. By using a generalized parametric vector companion form, the problem of eigen...
متن کاملFuzzy subgroups of the direct product of a generalized quaternion group and a cyclic group of any odd order
Bentea and Tu{a}rnu{a}uceanu~(An. c{S}tiinc{t}. Univ. Al. I.Cuza Iac{s}, Ser. Nouv{a}, Mat., {bf 54(1)} (2008), 209-220)proposed the following problem: Find an explicit formula for thenumber of fuzzy subgroups of a finite hamiltonian group of type$Q_8times mathbb{Z}_n$ where $Q_8$ is the quaternion group oforder $8$ and $n$ is an arbitrary odd integer. In this paper weconsider more general grou...
متن کاملOn the Error Parameter of Dispersers
Optimal dispersers have better dependence on the error than optimal extractors. In this paper we give explicit disperser constructions that beat the best possible extractors in some parameters. Our constructions are not strong, but we show that having such explicit strong constructions implies a solution to the Ramsey graph construction problem.
متن کاملAn E icient Reduction from Two-Source to Non-malleable Extractors
The breakthrough result of Chattopadhyay and Zuckerman (2016) gives a reduction from the construction of explicit two-source extractors to the construction of explicit non-malleable extractors. However, even assuming the existence of optimal explicit nonmalleable extractors only gives a two-source extractor (or a Ramsey graph) for poly(logn) entropy, rather than the optimal O (logn). In this pa...
متن کاملA Note on Explicit Ramsey Graphs and Modular Sieves
In a previous work [4] we found a relation between the ranks of codiagonal matrices (matrices with 0's in their diagonal and non-zeroes elsewhere) and the quality of explicit Ramsey-graph constructions. We also gave there a construction based on the BBR-polynomial of Barrington, Beigel and Rudich [1]. In the present work we give another construction for low-rank co-diagonal matrices, based on a...
متن کامل