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的存在性，而且这个值非常接近0.0185067，比1/54小一点，这是一个明显的构造。这一结果被Combinatorica所接受，如果任意r条边至少有t个公共顶点，则称子集族为r-向t-相交子集族。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-wise 2-相交Sperner族的最大尺寸。然后，我们转向了3明智的情况，这比4明智的情况要困难得多。利用随机游动方法，最终得到了3-wise 2-相交Spernar族的Erdos-Ko-Rado型不等式，从而成功地确定了3-wise 2-相交Spernar族的最大规模.我们已经写了两篇关于这个主题的论文，一篇发表在J.Comb.Theory上，另一篇已经提交到同一杂志上，我在ZiF研究年，开幕会议“General Theory of Information Transfer and Combinatorics”，ISM Symposium“Statistics，Combinatorics and Geometry”和Renyi Institute“Workshop on Extremal Combinatorics”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."