Weak Saturation of Multipartite Hypergraphs

Abstract Given q -uniform hypergraphs ( -graphs) F , G and H where is a spanning subgraph of called weakly - saturated in if the edges $$E(F)\setminus E(G)$$ E ( F ) \ G admit an ordering $$e_1,\ldots e_k$$ e 1 , … k so that for all $$i\in [k]$$ i ∈ [ ] hypergraph $$G\cup \{e_1,\ldots ,e_i\}$$ ∪ { } contains isomorphic copy which turn edge $$e_i$$ . The weak saturation number smallest size -weakly Weak was introduced by Bollobás 1968, but despite decades study our understanding it still limited. main difficulty lies proving lower bounds on numbers, typically withstands combinatorial methods requires arguments algebraic or geometrical nature. In contribution this paper we determine exactly complete multipartite -graphs directed setting, any choice parameters. This generalizes theorem Alon from 1985. Our proof combines exterior algebra approach works Kalai with use colorful motivated recent work Bulavka, Goodarzi Tancer fractional Helly theorem. second answering question Kronenberg, Martins Morrison, establish link between numbers bipartite graphs clique versus host graph. similar fashion asymptotically -partite -graph clique, generalizing another result Kronenberg et al.