In this paper, we will discuss some upper bound formulas for Ramsey numbers.At first, we will derive a more unified form from two parameter formulas which was obtained by Yiru Huang etc. Then, similar to their methods, we introduce two new bound formulas for Ramsey numbers in this paper. At last, as the applications of our new bound formulas, some upper bounds for small Ramsey numbers will be obtained.

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Let p be a prime and L be a finite extension of ℚp. We study the ordinary parts of GL2(L)-representations arised in the mod p cohomology of Shimura curves attached to indefinite division algebras which splits at a finite place above p. The main tool of the proof is a theorem of Emerton \cite{Em3}.