TY - JOUR

T1 - Weak Saturation of Multipartite Hypergraphs

AU - Bulavka, Denys

AU - Tancer, Martin

AU - Tyomkyn, Mykhaylo

N1 - Publisher Copyright:
© 2023, The Author(s).

PY - 2023

Y1 - 2023

N2 - 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, … , ek 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.

AB - 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, … , ek 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.

KW - Exterior algebra

KW - Hypergraph

KW - Saturation

UR - http://www.scopus.com/inward/record.url?scp=85165868854&partnerID=8YFLogxK

U2 - 10.1007/s00493-023-00049-0

DO - 10.1007/s00493-023-00049-0

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AN - SCOPUS:85165868854

SN - 0209-9683

JO - Combinatorica

JF - Combinatorica

ER -