标题:
高维Ramsey数问题On High-Dimensional Ramsey Number
作者:
单传辉
关键字:
概率方法, 高维Ramsey数理论, 着色Probabilistic Method, High-Dimensional Ramsey Number Theory, Coloring
期刊名称:
《Pure Mathematics》, Vol.5 No.5, 2015-09-11
摘要:
Paul Erdös和Noga Alon等人给出了一般意义的Ramsey数理论，本文通过概率方法给出了高维情况下的Ramsey数理论及其推广的一般形式。给出了等概率2-着色情况：当k=l时Rd(k,k)的下界结果；当k与l有区分时Rd(l,k)的下界结果。给出了等概率3-着色情况：当k=l=m时Rd(k,k,k)的下界结果；当k,l,m有区分时Rd(l,k,m)的下界结果。给出了r-着色情况下：当k1=k2=...=kr时Rd(k,k,...k)的下界结果；当k1,k2,...,kr有区分时Rd(k1,k2,...,kr)的下界结果。最后又给出了不等概率时上述各种情况下的相应结果。
Paul Erdös and Noga Alon give a general sense of Ramsey Number Theory. This paper presents a general form of Ramsey Number Theory in the dimensional case and its promotion by using probabilistic methods. The paper shows the lower bound of Rd(k,k) in the case of k=l with equal probabilistic 2-coloring situation, and shows the lower bound of Rd(l,k) in the case of k and l distinguishing with equal probabilistic 2-coloring situation. The paper shows the lower bound of Rd(k,k,k) in the case of k=l=m with equal probabilistic 3-coloring situation, and shows the lower bound of Rd(l,k,m) in the case of k, l and m distinguishing with equal probabilistic 3-coloring situation. And the paper shows the lower bound of Rd(k,k,...k) in the case of k1=k2=...=kr with equal probabilistic r-coloring situation, and shows the lower bound of Rd(k1,k2,...,kr) in the case of k1, k2,...,kr distinguishing with equal probabilistic r-coloring situation. Finally, we also give the corresponding results of the variety of situations with unequal probabilities.