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On Frankl and Füredi’s conjecture for 3-uniform hypergraphs

On Frankl and Füredi’s conjecture for 3-uniform hypergraphs 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 $$ \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right) - 6 \leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right)$$ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Acta Mathematicae Applicatae Sinica Springer Journals

On Frankl and Füredi’s conjecture for 3-uniform hypergraphs

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Publisher
Springer Journals
Copyright
Copyright © 2016 by Institute of Applied Mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences and Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Applications of Mathematics; Math Applications in Computer Science; Theoretical, Mathematical and Computational Physics
ISSN
0168-9673
eISSN
1618-3932
DOI
10.1007/s10255-015-0513-1
Publisher site
See Article on Publisher Site

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 $$ \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right) - 6 \leqslant m \leqslant \left( {\begin{array}{*{20}{c}}t \\ 3\end{array}} \right)$$ .

Journal

Acta Mathematicae Applicatae SinicaSpringer Journals

Published: Apr 5, 2016

References