2018/06/29 by Lei, Hui, Lu, Linyuan
#05C65 #05D05 #Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1806.11259
Frankl and Füredi conjectured in 1989 that the maximum Lagrangian, denoted by λr(m), among all r-uniform hypergraphs of fixed size m is achieved by the minimum hypergraph Cr,m under the colexicographic order. We say m in \em principal domain if there exists an integer t such that t-1\choose r≤ m≤ t\choose r-t-2\choose r-2. If m is in the principal domain, then Frankl-Füredi's conjecture has a very simple expression: λr(m)=(1)/((t-1)r)t-1\choose r. Many previous results are focusing on r=3. For r≥ 4, Tyomkyn in 2017 proved that Frankl-Füredi's conjecture holds whenever t-1\choose r ≤ m ≤ t\choose r -t-2\choose r-2- δrtr-2 for a constant δr>0. In this paper, we improve Tyomkyn's result by showing Frankl-Füredi's conjecture holds whenever t-1\choose r ≤ m ≤ t\choose r -t-2\choose r-2- δr'tr-(7)/(3) for a constant δr'>0.