Dirac-type problems for hypergraphs

尽管组合学与人类的计数能力一样古老，但它最令人印象深刻的增长是在最近几十年才出现的，这主要是由于计算机科学的快速发展。极值组合学研究在一类有限对象上求函数的最大值或最小值的问题。这些问题通常与数学的其他分支和其他科学领域有关，包括生物学、化学、计算机科学、信息和编码理论。本文主要研究超图的极值问题。由于超图具有比图更复杂的结构，因此将图的结果推广到超图通常远非简单，并且经常会产生新的现象。例如，有多种方法（取决于如何定义超图中的度和圈）可以将Dirac关于Hamilton圈的经典结果扩展到超图中——它们都不容易证明。狄拉克问题以及与之相关的匹配和包装问题近年来受到了广泛的关注。现代工具，如吸收法和正则法，有助于产生新的结果，但该领域的许多基本问题仍未得到解决。PI计划解决该领域的许多突出问题，发展超图的极值理论。

Although Combinatorics is as old as human's ability to count, its most impressive growth has been seen only in recent decades, mainly due to the rapid development of computer science. Extremal Combinatorics deals with the problems of finding the maximum or minimum value of a function over a class of finite objects. Such problems are often related to other branches of mathematics and other fields of science, including biology, chemistry, computer science, information and coding theory. The proposed research concentrates on extremal problems for hypergraphs. Since hypergraphs have more complicated structures than graphs, generalizing results from graphs to hypergraphs is usually far from straightforward and frequently gives rise to new phenomena. For example, there are multiple ways (depending on how to define degrees and cycles in hypergraphs) to extend the classical result of Dirac on Hamilton cycles to hypergraphs -- none of them has an easy proof. The Dirac problems, and related matching and packing problems have received considerable attention lately. Modern tools, e.g., the absorbing method and the regularity method, have helped to generate new results, and yet many fundamental problems in the area remain unsolved. The PI plans to tackle many outstanding problems in the area, developing the extremal theory of hypergraphs.