A short proof of fisher's inequality

Discrete Mathematics North-Holland

111 (1993) 421-422

421

A short proof of Fisher’s inequality Renaud U.P.R.

Received

Pal&e,

Palisse

175 -

Universitl Pierre et Marie Curie

22 July 1991

R., A short proof of Fisher’s inequality,

Discrete

Mathematics

111 (1993) 421422

Nous donnons ici une dtmonstration nouvelle, trts courte, de I’iGgaliti de Fisher, qui gCntralise un rCsultat bien connu de de Bruijn et ErdGs. Cette dtmonstration utilise essentiellement une id&e de Tverberg (1982) pour dt-montrer un autre tnonct combinatoire.

We prove

the following

result.

a,> be a finite

Theorem. Let E=jal,,... n subsets of E such that

fori#j

IEinEjI=3.

and

set of cardinal&y

m and let El,

. . , E, be

1Eil > 1. for all i.

Then man. that j_#O. Let x1, . .., x, be real variables.

Proof. We may assume we may write

a,EEinEj)=J-

C

Since )Ein Ej/ = A,

XiXj,

i
Hence,

If we had m
akFE xi=o

(ldkdm)

Correspondence to: Renaud Africa. Elsevier Science Publishers

system

Palisse,

B.V.

Mission

Cooperation,

B.P. 510, Ovagadougou,

Burkina

Fasso,

422

R. Palisse

would have a nonzero

solution

(~ 12=~ Xi

i=l

a contradiction.

X and, consequently,

(l-IEil/A)%~
i=l

Thus, m > n.

0

For the usual proof of this inequality, For similar

methods

see [3].

of proof for other combinatorial

results, see [l, 2,4,5].

[I] Q. Huang, On the decomposition of K, into complete m-partite graphs, J. Graph Theory 15 (1991) 1-6. [2] D. Pritikin, Applying a proof of Tverberg to complete bipartite decompositions of digraphs and multigraphs, J. Graph Theory 10 (1986) 197-201. [3] H.J. Ryser, An extension of a theorem of de Bruijn and Erdiis on combinatorial designs, J. Algebra 10 (1968) 246-261. [4] P. Seymour, On 2-colourings of hypergraphs, Quart. J. Math. Oxford 25 (1974) 303-312. [5] H. Tverberg, On the decomposition of K, into complete bipartite graphs, J. Graph Theory 6 (1982) 4933494.