Proof of a conjecture of Frankl and Furedi

Ramanan, Gurumurthi V (1997) Proof of a conjecture of Frankl and Furedi. In: Journal of Combinatorial Theory - Series A, 79 (1). pp. 53-67.

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Abstract

We give a simple linear algebraic proof of the following conjecture of Frankl and Furedi [7, 9, 13]. (Frankl-Furedi Conjecture) if F is a hypergraph on X = {1, 2, 3,..., n} such that 1 less than or equal to /E boolean AND F/ less than or equal to k For All E, F is an element of F, E not equal F, then /F/ less than or equal to (i=0)Sigma(k) ((i) (n-1)). We generalise a method of Palisse and our proof-technique can be viewed as a variant of the technique used by Tverberg to prove a result of Graham and Pollak [10, 11, 14]. Our proof-technique is easily described. First, we derive an identity satisfied by a hypergraph F using its intersection properties. From this identity, we obtain a set of homogeneous linear equations. We then show that this defines the zero subspace of R-/F/. Finally, the desired bound on /F/ is obtained from the bound on the number of linearly independent equations. This proof-technique can also be used to prove a more general theorem (Theorem 2). We conclude by indicating how this technique can be generalised to uniform hypergraphs by proving the uniform Ray-Chaudhuri-Wilson theorem. (C) 1997 Academic Press.

Item Type:	Journal Article
Publication:	Journal of Combinatorial Theory - Series A
Publisher:	Elsevier Science
Additional Information:	Copyright of this article belongs to Elsevier Science.
Department/Centre:	Division of Physical & Mathematical Sciences > Mathematics
Date Deposited:	30 Jun 2011 13:54
Last Modified:	30 Jun 2011 13:54
URI:	http://eprints.iisc.ac.in/id/eprint/38373

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