 
Summary: JOURNAL OF COMBINATORIAL THEORY, Series A 43, 9197 (1986)
Regular Hypergraphs, Gordon's Lemma,
Steinitz' Lemma and Invariant Theory
N. ALON* AND K. A. BERMAN
Department of Mathematics, Massachusetts Institute of Technology,
Cambridge, Massachusetts 02139
Communicated by the Managing Editors
Received March 25, 1985
Let D(n)(D(n, k)) denote the maximum possible d such that there exists a
dregular hypergraph (dregular kuniform hypergraph, respectively) on n vertices
containing no proper regular spanning subhypergraph. The problem of estimating
D(n) arises in Game Theory and Huckemann and Jurkat were the first to prove
that it is finite. Here we give two new simple proofs that D(n), D(n, k) are finite,
and determine D(n, 2) precisely for all n > 2. We also apply this fact to Invariant
Theory by showing how it enables one to construct an explicit finite set of
generators for the invariants of decomposable forms. 0 1986 Academic Press, Inc.
1. INTRODUCTION
Suppose n > 1 and put N = ( 1,2, 3,..., n}. A (multi)hypergraph H on N is
a multiset of elements of the power set P(N), i.e., a collection of subsets of
N, where the same subset can appear several times. The degree of a point
