The Lagrangians Of Uniform Hypergraphs

In 1980's, Frankl and Furedi conjectured that if G is an r-uniform graph with m edges, then where ?(G) is the lagrangian of G, Cr,m is the r-uniform graph with m edges formed by taking the first m sets in N(r) in colex ordering. Talbot in [1] first confirmed Frankl and Furedi's conjecture for r=3 and Later, Tang et. al. extended Talbot's result to r= 3 and (3t-1) It seems to be challenging to confirm this conjecture even for r= 3. Peng and Zhao conjectured that if G is an r-graph with t vertices and m edges, with and then In [5], Peng and Zhao proved that this conjecture holds for r= 3. They also conjectured:Let G be a r-graph with t vertices, m edges and without containing a clique of order t - 1, where then The truth of this pair of conjectures will imply the truth of Frankl-Fiiredi's conjecture for this range of m. In this paper, we mainly focus on the second conjecture for r=3. Peng et. al. showed that to verify the above conjecture for r=3, it is sufficient to check for left-compressed 3-uniform graphs on [t] with edges. For conjecture 2, sun et al in [6] proved conjecture 2 for r= and |E(t-1)t|?7. In this thesis, we give some partial results to this conjecture when...