Abstract
Frankl and Füredi in [1] conjectured that the r-graph with m edges formed by taking the first m sets in the colex ordering of N(r) has the largest Lagrangian of all r-graphs with m edges. Denote this r-graph by C r,m and the Lagrangian of a hypergraph by λ(G). In this paper, we first show that if \(\leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3 \end{array}} \right)\), G is a left-compressed 3-graph with m edges and on vertex set [t], the triple with minimum colex ordering in G c is (t − 2 − i)(t − 2)t, then λ(G) ≤ λ(C 3,m ). As an implication, the conjecture of Frankl and Füredi is true for \( \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right) - 6 \leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right)\).
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Supported by Chinese Universities Scientic Fund (No.N140504004), the National Natural Science Foundation of China (No. 11271116).
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Tang, Qs., Peng, H., Wang, Cl. et al. On Frankl and Füredi’s conjecture for 3-uniform hypergraphs. Acta Math. Appl. Sin. Engl. Ser. 32, 95–112 (2016). https://doi.org/10.1007/s10255-015-0513-1
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DOI: https://doi.org/10.1007/s10255-015-0513-1