Application Of Probabilistic Method In Combination Mathematics

The application of probabilistic method in combination mathematics can be dividedinto two kinds: the first one is called non-structuring probabilistic method. Basicallyspeaking, this method is a simply rough counting and demonstrating way. Generally, thismethod is used to prove the existence of a certain feature combination. The second iscalled structuring Probability method. This method aims at using the probabilisticlanguage to describe some combination object, and then to solve the problems inCombinatorial analysis. One always uses the following ways to solve the combinatorialmathematical problem: first proving the existence of structure with some certainexpectation qualities, defining a probability space composed of these structures, and thenby the positive of the probability to prove the existence of these expectation qualities.This paper aims at studying some important applications of probabilistic method incombinatorial mathematics, which consists of five parts：The first chapter, we introduce some background of probabilistic method andcombinatorial mathematics.The second chapter is devoted to studying the application and promotion ofprobabilistic method in the Ramsey function, some theorem about the Ramsey functionand the proof by probabilistic method. Then estimate the upper and lower bounds of theRamsey function by probabilistic method.The third chapter is devoted to studying the application of probabilistic method inrandom graph, introduces the concept and theorem related to the random graph, and thenmainly introduce the method to solve the threshold function with certain quality in randomgraph.In the fourth chapter, we study the the application of probabilistic method incombinatorial number theory, introduce some basic concepts in combinatorial number theory, and then by the constructing probabilistic method, optimize the large number ofpartial, genetic differences, linear difference.The fifth chapter, mainly summarizes the applications of probabilistic methodnowadays and foresee the future development of probabilistic method in combinationmathematics.