Lagrangians of Hypergraphs

Abstract

How large can the Lagrangian of an r-graph with m edges be? Frankl and Füredi [1] conjectured that the r-graph of size m formed by taking the first m sets in the colex ordering of N(r) has the largest Lagrangian of all r-graphs of size m. We prove the first ‘interesting’ case of this conjecture, namely that the 3-graph with (t3) edges and largest Lagrangian is [t](3). We also prove that this conjecture is true for 3-graphs of several other sizes.For general r-graphs we prove a weaker result: for t sufficiently large, the r-graph of size (tr) supported on t + 1 vertices and with largest Lagrangian, is [t](r).

Lagrangians of hypergraphs: The Frankl-Füredi conjecture holds almost everywhere

“…x i = 1 and L(G, x) = λ(G) (note that in general G and x are not unique). Following the conventional notation (see, for example, [8]), we can assume by symmetry that the entries of x are listed in descending order, that is x i x j for all i < j. We shall furthermore assume that, subject to the above conditions, x has the minimum possible number of non-zero entries, and let T be this number.…”

“…The next statement holds in more generality, but we shall mainly need it for G and x. Proposition 2.8 [4,8]. Let G, T and x be as defined above.…”

“…As our induction base we take Corollary 1.4 for r = 3, which is known to hold by Talbot's theorem [8]. Note though, that our proof does not crucially rely on [8], as, alternatively, we could start at r = 3, taking the trivial r = 2 case as the induction base; this would also give a new proof of a slightly weaker form of Talbot's theorem for r = 3.…”

“…In contrast to this, for r 4 much less has been known so far, as Talbot's proof method for r = 3, perhaps surprisingly, does not immediately transfer. Talbot showed in the same paper [8] that for every r 4 there is a constant γ r > 0 such that if t−1 r m t r − γ r t r−2 and H is supported on t vertices (that is, ignoring isolated vertices, H is a subgraph of [t] (r) ), then indeed λ(H) λ(H m,r ). Still, for no value of m, apart from some trivial ones, Conjecture 1.1 has been known to hold.…”

“…, that is for all m ∈ N, the 3-graph maximising the Lagrangian amongst all m-edge 3-graphs can be assumed to be supported on [t]. Combined with Theorem 1.5, this yields, for large m, an improvement of the bounds in [8] and [9].…”