Given q-uniform hypergraphs (q-graphs) F, G and H, where G is a spanning subgraph of F, G is called weakly H-saturated in F if the edges in \(E(F)\setminus E(G)\) admit an ordering \(e_1,\ldots , e_k\) so that for all \(i\in [k]\) the hypergraph \(G\cup \{e_1,\ldots ,e_i\}\) contains an isomorphic copy of H which in turn contains the edge \(e_i\). The weak saturation number of H in F is the smallest size of an H-weakly saturated subgraph of F. Weak saturation was introduced by Bollobás in 1968, but despite decades of study our understanding of it is still limited. The main difficulty lies in proving lower bounds on weak saturation numbers, which typically withstands combinatorial methods and requires arguments of algebraic or geometrical nature. In our main contribution in this paper we determine exactly the weak saturation number of complete multipartite q-graphs in the directed setting, for any choice of parameters. This generalizes a theorem of Alon from 1985. Our proof combines the exterior algebra approach from the works of Kalai with the use of the colorful exterior algebra motivated by the recent work of Bulavka, Goodarzi and Tancer on the colorful fractional Helly theorem. In our second contribution answering a question of Kronenberg, Martins and Morrison, we establish a link between weak saturation numbers of bipartite graphs in the clique versus in a complete bipartite host graph. In a similar fashion we asymptotically determine the weak saturation number of any complete q-partite q-graph in the clique, generalizing another result of Kronenberg et al.

给定q -均匀超图 ( q -图) F、  G和H，其中G是F的生成子图，如果\(E(F)\setminus E(G)中的边在F中，则G被称为弱 H -饱和\)承认一个排序\(e_1,\ldots , e_k\)以便对于所有\(i\in [k]\)来说，超图\(G\cup \{e_1,\ldots ,e_i\}\)包含一个H的同构副本，其中又包含边\(e_i\)。弱饱和数F中的H是F的H弱饱和子图的最小尺寸。弱饱和度由 Bollobás 于 1968 年提出，但尽管经过数十年的研究，我们对它的理解仍然有限。主要困难在于证明弱饱和数的下界，该下界通常可以承受组合方法并且需要代数或几何性质的论证。在本文的主要贡献中，我们准确地确定了完整多部分q的弱饱和数-针对任何参数选择，在定向设置中绘制图表。这概括了 1985 年的 Alon 定理。我们的证明结合了 Kalai 作品中的外代数方法和由 Bulavka、Goodarzi 和 Tancer 最近关于彩色分数 Helly 定理的工作所激发的彩色外代数的使用。在回答 Kronenberg、Martins 和 Morrison 问题的第二篇贡献中，我们在派系中的二分图的弱饱和数与完整二分主图中的弱饱和数之间建立了联系。以类似的方式，我们渐进地确定团中任何完整q分q图的弱饱和数，推广了 Kronenberg 等人的另一个结果。