Weak saturation of multipartite hypergraphs

TL;DR

This paper precisely determines the weak saturation number of complete multipartite hypergraphs in directed settings, extending classical results and introducing algebraic methods, while also linking bipartite graph saturation in different host graphs.

Contribution

It provides exact formulas for weak saturation numbers of complete multipartite hypergraphs and establishes new connections between saturation in bipartite graphs within different host graphs.

Findings

Exact weak saturation number for complete multipartite hypergraphs in directed setting.

Asymptotic determination of weak saturation numbers in complete bipartite graphs.

Extension of classical theorems using algebraic and geometric methods.

Abstract

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 $e_1,\dots, e_k$ so that for all $i\in [k]$ the hypergraph $G\cup \{e_1,\dots,e_i\}$ 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 Bollob\'as 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.