On Combinatorial Congruences And Additive Combinatorics

Combinatorial congruences and additive combinatorics are important parts of Combinatorial Number Theory.In this thesis we study combinatorial congruences of Ramanujan type and extensions of the solved Erdos-Heilbronn conjecture on sumset with distinct summands.Combinatorial congruences are related to many fields of mathematics,such as p-adic analysis and Ramanujan-type series for 1/?.In this thesis,by using tools of Wilf-Zeilberger pairs,combinatorial identities and Bernoulli numbers,we prove the following two congruences conjectured by Z.W.Sun:For any prime p>3 we have and We also make progress towards another congruence conjectured by Z.W.Sun.Additive Combinatorics is concerned with combinatorial properties related to ad-ditive structure.This field is quite active in recent years.In this thesis,we mainly apply the polynomial method based on Alon's Combinatorial Nullstellenstaz to obtain a polynomial extension of the solved Erdos-Heilbronn conjecture.This thesis consists of five chapters.In the first chapter,we first present a survey of supercongruences related to Ramanujan-type series for 1/?,and introduce known extensions of the Erdos-Heilbronn conjecture,and then we state our main results.Chapters 2-5 are devoted to our proofs of the main results in this thesis.