An extension of Maclaurin's inequalities

Localised graph Maclaurin inequalities

The Maclaurin inequalities for graphs are a broad generalisation of the classical theorems of Turán and Zykov. In a nutshell they provide an asymptotically sharp answer to the following question: what is the maximum number of cliques of size $q$ in a $K_{r+1}$-free graph with a given number of cliques of size $s$? We prove an extensions of the graph Maclaurin inequalities with a weight function that captures the local structure of the graph. As a corollary, we settle a recent conjecture of Kirsch and Nir, which simultaneously encompass the previous localised results of Bradač, Malec and Tompki…

有限偏序集线性扩张数的相关不等式

Correlation inequalities for linear extensions

该研究采用组合图谱技术证明有限偏序集线性扩张数的新相关不等式（含与Stanley不等式相关的近似独立性结果），并拓展应用至标准杨表数和欧拉数领域。

有限域上矩曲线的尖锐端点延拓不等式

Sharp endpoint extension inequalities for the moment curve on finite fields

本文研究了有限域中矩曲线的尖锐端点延拓不等式，通过结合分析、代数与组合学方法，确定了最优常数，并在低维与大基域基数两种互补情形下刻画了最大化子。

Extension results for boolean maps and a class of systems of linear inequalities

In this paper we introduce the notion of {\it core} for two specific classes of boolean maps on finite involution posets (which are a generalization of the boolean lattices) and we prove some extension results for such families of boolean maps. Through the properties of the core, we provide a complete characterization of such maps. The main purpose of such abstract results is their application to the study of the compatibility of a particular class of systems of linear inequalities related to a conjecture of Manickam, Miklös and Singhi (\cite{ManSin88}, \cite{ManMik87}), still unsolved and tha…

有限偏序集的线性扩张

Linear extensions of finite posets

本文综述了有限偏序集线性扩张数量的不等式，涵盖典型例子、未解决问题、最新结果、界与等号条件及计算复杂性。

相关不等式的等式条件

Equality conditions for correlation inequalities

研究相关不等式的等式条件，证明了AD不等式和FKG不等式的等式条件，给出其应用与特殊情况，还得出ADS不等式的等式条件，ADS不等式是LPP不等式的AD型扩展。