Research on the minimum ai'ea of convex lattice polygons and multiply intersecting families

设A(n)为凸格n-gon的最小面积。我们已经证明了lim A(n)/n^3的存在，并且这个值非常接近于0.0185067，略小于1/54，这是一个明显的构造。这个结果被Combinatorica接受。如果任意r条边至少有t个公共顶点，则一组子集称为r向t相交。如果子集族之间不存在包含关系，则称其为Sperner子集族。我们已经确定了4- 2相交斯伯纳族的最大大小。然后我们开始讨论3明智的情况，这比4明智的情况要困难得多。利用随机漫步法，最终得到了3-wise 2- intersection族的Erdos-Ko-Rado型不等式，进而成功确定了3-wise 2- intersection Spernar族的最大大小。关于这个话题，我们已经写了两篇论文，一篇发表在J.Comb杂志上。理论上，另一篇论文也投到了同一杂志。我在ZiF研究年度、“信息传递与组合通论”开幕会议、ISM“统计、组合与几何”研讨会和人艺研究所“极值组合研讨会”上做了三次演讲。

Let A(n) be the-minimum area of convex Lattice n-gon. We have proved that lim A(n)/n^3 exists, and moreover this value is very close to 0.0185067, which is a little bit less than 1/54 which comes from an obvious construction. This result is accepted by Combinatorica.A family of subsets is called r-wise t-intersecting if any r edges have at least t common vertices. A family of subsets is called Sperner if it contains no inclusion relationship among subsets. We have determined the maximum size of 4-wise 2-intersecting Sperner families. Then we moved on 3-wise case, which was much more-difficult than 4-wise case. Using random walk method, we finally obtained Erdos-Ko-Rado type inequality for 3-wise 2-intersecting families, and then we succeeded to determine the maximum size of 3-wise 2-intersecting Spernar families. We have wrote two papers on this topic, one appears in J.Comb.Theory, the other has been submitted to the same journal.I did three lectures about the above results as an invited talk at ZiF research year, opening conference "General theory of information transfer and combinatorics", ISM Symposium "Statistics, Combinatorics and Geometry" and Renyi Institute "Workshop on extremal combinatorics."