Combinatorial inequalities
Author
Qian, Jin
Year
2000
Advisor
Ray-Chaudhuri, Dijen K.
Abstract
First in Chapter 2 we generalize the concept of a polynomial semilattice introduced by Ray-Chaudhuri and Zhu [23] as a general framework in which to study the theory of combinatorial inequalities, We generalize it to a quasi polynomial semilattice and then show by example that it strictly contains the polynomial semilattice as a special case. Also in Chapter 2 we show many of the classic combinatorial inequalities hold in the context of quasi polynomial semilattice. Secondly, the extreme case of Franlyn-Ray-Chaudhuri-Wilson inequality is classified. It is shown in Theorem 3.2.1 of Chapter 3 that the there is only one family that can attain the upper bound given in that inequity. Thirdly, a theorem is proved in Chapter 4 that confirms a conjecture of Alon, Babai and Suzuki to a large extent. Finally, in chapter 5 we give proofs to some special cases of Snevily's conjecture and prove a theorem which shows that under a mild condition there is only one family that can attain the upper bound in the Alon-Babai-Suzuki inequality.
Qian, Jin
Year
2000
Advisor
Ray-Chaudhuri, Dijen K.
Abstract
First in Chapter 2 we generalize the concept of a polynomial semilattice introduced by Ray-Chaudhuri and Zhu [23] as a general framework in which to study the theory of combinatorial inequalities, We generalize it to a quasi polynomial semilattice and then show by example that it strictly contains the polynomial semilattice as a special case. Also in Chapter 2 we show many of the classic combinatorial inequalities hold in the context of quasi polynomial semilattice. Secondly, the extreme case of Franlyn-Ray-Chaudhuri-Wilson inequality is classified. It is shown in Theorem 3.2.1 of Chapter 3 that the there is only one family that can attain the upper bound given in that inequity. Thirdly, a theorem is proved in Chapter 4 that confirms a conjecture of Alon, Babai and Suzuki to a large extent. Finally, in chapter 5 we give proofs to some special cases of Snevily's conjecture and prove a theorem which shows that under a mild condition there is only one family that can attain the upper bound in the Alon-Babai-Suzuki inequality.