On Lagrangians of 3-uniform hypergraphs

2018/06/28 by Lei, Hui, Lu, Linyuan, Peng, Yuejian
#05C65 #05D05 #Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1806.10846

Abstract

Frankl and Füredi conjectured in 1989 that the maximum Lagrangian of all r-uniform hypergraphs of fixed size m is realized by the minimum hypergraph C_r,m under the colexicographic order. In this paper, we prove a weaker version of the Frankl and Füredi's conjecture at r=3: there exists an absolute constant c>0 such that for any 3-uniform hypergraph H with m edges, the Lagrangian of H satisfies λ(H)≤ λ(C_3,m+cm^2/9). In particular, this result implies that the Frankl and Füredi's conjecture holds for r=3 and m∈ [t-1\choose 3, t\choose 3-(t-2)-ct^(2)/(3)]. It improves a recent result of Tyomkyn.

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