Weak saturation of multipartite hypergraphs

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 ) \ E(G) admit an ordering e1, . . . , e k so that for all i ∈ [k] the hypergraph G ∪ {e1, . . . , ei} contains an isomorphic copy of H which in turn contains the edge ei. 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.

Weakly saturated hypergraphs and a conjecture of Tuza

“…While the aforementioned results determine the exact value of wsat(n, H) when H = K r t , our understanding of this function for general H is much more limited, despite decades of extensive study [1,3,4,9,14,17,25,26,29,30,35,36,38,39]. Note that by the construction from [15], we know that every graph H we have…”