极值组合学中的相交族问题及其应用

相交族是极值组合学的一个重要概念，它起源于著名的Erdős-Ko-Rado定理，与图论、结合方案、离散几何、编码和设计等领域有着紧密联系。在本项目中，我们将采用组合、代数、几何等方法，利用t-覆盖数、结合方案、矩阵几何等工具，研究给定覆盖数的1-相交族、基数最大的非平凡t-相交族和交叉t-相交族，并讨论它们在图论、编码和pooling设计等领域中的应用。

Intersecting family is an important concept in extremal combinatorics. It originates from the famous Erdős-Ko-Rado theorem, and is closely related to graph theory, association schemes, discrete geometry, coding theory and design theory. In this project, applying combinatorial, algebraic and geometric methods, and using tools such as t-covering number, association scheme and matrix geometry, we study the 1-intersecting families with fixed covering number, non-trivial t-intersecting families with maximum cardinality and cross t-intersecting families. We also study their applications in graph theory, coding theory and pooling design.