An equation for the general Ramsey number R(p1,p2,...pt;r)

TL;DR

This paper derives an equation for the general Ramsey number R(p1,p2,...,pt;r) by analyzing the distribution of r-subsets in an n-set and applying elementary counting methods.

Contribution

It introduces a novel equation for calculating the general Ramsey number R(p1,p2,...,pt;r) based on combinatorial distribution analysis.

Findings

Provides a formula for the general Ramsey number R(p1,p2,...,pt;r)

Reduces the problem to elementary counting methods

Offers a new approach to evaluate Ramsey numbers

Abstract

The Ramsey number $R(P_1,P_2,...,P_t;r)$ is a valve value such that as long as the cardinality $n$ of the $n$-set $V_n={1,...,n}$ is no less than $R$,however all the $\binom{n}{r}$ $r$-subsets of $V_n$ are distributed into $t$ boxes, $V_n$ will always have a property $W$ expressed as eq.(1).Thus, by calculating the number of ways of distribution of $r$-subsets that makes $W$ true,one can get an equation for $R(P_1,P_2,...,P_t;r)$.The evaluation of the general term in this eq. and the counting of the frequencies of occurrence of the various values the general term takes can be reduced to the problem of elementary counting.