2018/10/31 by Peng, Yuejian, Yan, Zilong
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1810.13077
The Lagrangian density of an r-uniform hypergraph F is r! multiplying the supremum of the Lagrangians of all F-free r-uniform hypergraphs. For an r-graph H with t vertices, it is clear that πλ(H)≥ r!λ(Kt-1r). We say that an r-unform hypergraph H with t vertices is perfect if πλ(H)= r!λ(Kt-1r). A theorem of Motzkin-Straus implies that all 2-uniform graphs are perfect. It is interesting to explore what kind of hypergraphs are perfect. A hypergraph is linear if any 2 edges have at most 1 vertex in common. We propose the following conjecture: (1) For r≥ 3, there exists n such that a linear r-unofrm hypergraph with at least n vertices is perfect. (2) For r≥ 3, there exists n such that if G, H are perfect r-uniform hypergraphs with at least n vertices, then G\bigsqcup H is perfect. Regarding this conjecture, we obtain a partial result: Let S2,t=\123,124,125,126,...,12(t+2)\. (An earlier result of Sidorenko states that S2,t is perfect \citeSidorenko-89.) Let H be a perfect 3-graph with s vertices. Then F=S2,t\bigsqcup H is perfect if s≥ 3 and t≥ 3.