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For bipartite graphs , the bipartite Ramsey number is the least positive integer so that any coloring of the edges of with colors will result in a copy of in the th color for some . In this paper, our main focus will be to bound the following numbers: and for all for and for Furthermore, we will also show that these mentioned bounds are generally better than the bounds obtained by using the best known Zarankiewicz-type result. 相似文献
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Anuj Jakhar Sudesh K. Khanduja Neeraj Sangwan 《Journal of Pure and Applied Algebra》2018,222(11):3560-3565
It is well known that if are algebraic number fields with coprime discriminants, then the composite ring is integrally closed and are linearly disjoint over the field of rationals, being the ring of algebraic integers of . In an attempt to prove the converse of the above result, in this paper we prove that if are finite separable extensions of a valued field of arbitrary rank which are linearly disjoint over and if the integral closure of the valuation ring of v in is a free -module for with integrally closed, then the discriminant of either or of is the unit ideal. We quickly deduce from this result that for algebraic number fields linearly disjoint over for which is integrally closed, the relative discriminants of and must be coprime. 相似文献
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TextFor any given two positive integers and , and any set A of nonnegative integers, let denote the number of solutions of the equation with . In this paper, we determine all pairs of positive integers for which there exists a set such that for all . We also pose several problems for further research.VideoFor a video summary of this paper, please click here or visit http://www.youtube.com/watch?v=EnezEsJl0OY. 相似文献
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Kiyoshi Ando 《Discrete Mathematics》2018,341(11):3003-3009
An edge of a -connected graph is said to be -contractible if the contraction of the edge results in a -connected graph. If every -connected graph with no -contractible edge has either or as a subgraph, then an unordered pair of graphs is said to be a forbidden pair for -contractible edges. We prove that is a forbidden pair for 6-contractible edges, which is an extension of a previous result due to Ando and Kawarabayashi. 相似文献
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Let be a finite field of odd order and , where are positive integers, are distinct odd primes and . In this paper, we study the irreducible factorization of over and all primitive idempotents in the ring .Moreover, we obtain the dimensions and the minimum Hamming distances of all irreducible cyclic codes of length over . 相似文献
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Neha Hooda 《Journal of Pure and Applied Algebra》2018,222(10):3043-3057
Let k be a field of characteristic different from 2 and 3. In this paper we study connected simple algebraic groups of type , and defined over k, via their rank-2 k-tori. Simple, simply connected groups of type play a pivotal role in the study of exceptional groups and this aspect is brought out by the results in this paper. We refer to tori, which are maximal tori of type groups, as unitary tori. We discuss conditions necessary for a rank-2 unitary k-torus to embed in simple k-groups of type , and in terms of the mod-2 Galois cohomological invariants attached with these groups. The results in this paper and our earlier paper ([6]) show that the mod-2 invariants of groups of type and are controlled by their k-subgroups of type and as well as the unitary k-tori embedded in them. 相似文献