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

设A(N)是凸格n边形的-最小面积。我们已经证明了Lim A(N)/n^3的存在，而且这个值非常接近0.0185067，略小于一个明显结构的1/54。如果任何r条边至少有t个公共顶点，则称一个子集族为r向t-交。一个子集族称为Sperner，如果它不包含子集之间的包含关系。我们确定了4向2交Sperner族的最大尺寸。然后我们继续3-WISE案例，这比4-WISE案例要难得多。利用随机游动的方法，我们最终得到了三向二交Spernar族的Erdos-Ko-Rado型不等式，并成功地确定了三向二交Spernar族的最大规模。我们已经就这一主题撰写了两篇论文，一篇发表在《组合理论》杂志上，另一篇提交给了同一期刊。我作为ZIF研究年的特邀演讲，在开幕式上发表了三次关于上述结果的讲座，并在ISM研讨会上发表了《统计学、组合学和几何学》，仁义研究所举办了《极值组合学研讨会》。

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."