Extremal Combinatorics

我的研究将集中在极值组合。 极值集理论中的一个典型问题是，给定一个整数m和一个性质P，求{1，2，.}的子集族的最大尺寸，m}，其满足性质P。还可以寻求结构或枚举信息。Erdos、Stone和Simonovits的极图结果提供了例子。本文研究了无子图H的m个顶点的（简单）图G的边数的渐近界。渐近界是基于H的色数。有许多方法可以推广到集合系统，其中我们可以将集合解释为边。我们说一个（0，1）-矩阵是简单的，如果它没有重复的列。我们编码一个集系统作为一个简单的（0，1）-矩阵使用顶点集关联矩阵A。 子图一般化如下。设F是rxs（0，1）-矩阵. 我们说F是A的配置，如果存在A的rxs子矩阵，它是F的行和列置换。 我们考虑禁用配置。一个重要的例子是Kk，所有（0，1）列和k行的kx 2k矩阵。Vapnik和Chervonenkis为计算学习理论研究了这一点，它仍然很重要。 VC维是k的最大值，使得A包含Kk的副本。** Anstee和Sali指出，在一个简单的m行矩阵A中，没有配置F的最大列数可以从乘积构造的某个有限范围中渐近地确定（作为m的函数）。 我们将继续这个推测。****** 证明技术包括线性代数/多项式方法，巧妙的归纳，稳定性结果，移位，遗传算法和计算机辅助案例分析。 相关的问题包括禁止模式，Sperner理论，极值图论，Ramsey理论，区组设计，覆盖数组和区组设计。与萨利，一些最好的可能的界限时，发现禁止配置F的大小是一个函数的m。针对Anstee，Frankl，Furedi和Pach关于禁子矩阵的猜想，我们还研究了禁子矩阵的性质。使用摊销分析（来自计算机科学）。 ** 我也将研究图论中的问题。 一个示例问题是2 mx2 m棋盘。我们能用多米诺骨牌来覆盖它吗？这些对应于网格图中的完美匹配。 如果我们删除一些方块会发生什么？或者我们把多米诺骨牌的位置固定一下？ 这对更高维度的网格图有自然的扩展，其中2 mx2 mx2 m可以用来模拟分子结构。 我们发现了顶点删除的渐近最佳可能结果。 一个相关的问题是将图的边分解成具有某些性质的子图。将一个图分解成匹配就是边着色问题。 我利用的想法，寻找分数子图（即，允许边被选择与分数权重）是一个网络流问题。这产生了快速算法和存在定理以及有待探索的新观点。

My research will focus on Extremal Combinatorics. A typical problem in Extremal Set Theory is, given an integer m and a property P, to find the largest size of a family of subsets of {1,2,..,m} which satisfy property P. One can also seek structural or enumerative information. The remarkable extremal graph results of Erdos, Stone and Simonovits provide examples. One seeks asymptotic bounds on the number of edges in a (simple) graph G on m vertices with no subgraph H. The asymptotic bounds are based on the chromatic number of H. There are a number of ways to generalize to set system, where we can interpret the sets as edges. We say a (0,1)-matrix is simple if it has no repeated columns.We encode a set system as a simple (0,1)-matrix using the vertex-set incidence matrix A. A subgraph generalizes as follows. Let F be an rxs (0,1)-matrix. We say F is a configuration of A if there is an rxs submatrix of A which is a row and column permutation of F. We consider forbidden configurations. An important example is Kk, the kx2k matrix of all (0,1)-columns with k rows. Vapnik and Chervonenkis studied this for Computational Learning Theory and it remains important. The VC-dimension is the largest value of k so that A contains a copy of Kk. ****** Anstee and Sali conjectured that the maximum number of columns in a simple m-rowed matrix A that has no configuration F can be determined asymptotically (as a function of m) from a certain restricted range of product constructions. We will pursue this conjecture. ****** Proof techniques include linear algebra/polynomial methods, clever inductions, Stability results, Shifting, Genetic algorithms and computer aided case analysis. Related problems include forbidden patterns, Sperner Theory, extremal graph theory, Ramsey Theory, Block designs, covering arrays, and block designs.******With Sali, some best possible bounds were found when forbidding a configuration F whose size is a function of m. We also consider the property of a forbidden submatrix aiming for the conjecture of Anstee, Frankl, Furedi and Pach on Forbidden submatrices. Amortized analysis (from Computer Science) is used. ******I will also investigate problems in Graph Theory. A sample problem is the 2mx2m checkerboard. Can we cover it in dominoes? These correspond to perfect matchings in the grid graph. What happens if we delete some squares? Or if we fix the placement of some dominoes? This has natural extensions to grid graphs in higher dimension where the 2mx2mx2m can serve to model molecular structure. We found asymptotically best possible results for vertex deletion. A related problem is decomposing the edges of a graph into subgraphs with certain properties.***Decomposing a graph into matchings is the edge colouring problem. I exploit the idea that looking for fractional subgraphs (that is, allowing an edge to be chosen with a fractional weight) is a network flow problem. This yields fast algorithms and existence theorems and new perspectives to be explored.