Balanced extensions of graphs and hypergraphs

For a hypergraph G with v vertices and e~ edges of size i, the average vertex degree is d(G) = =Eiedv. Call balanced if d(H)~-d(G) for all subhyoergraphs H of G. Let re(G0= max d(H). Hc=G A hypergraph F is said to be a balanced extension of G if GC F, F is balanced and tifF)-re(G), i.e. F is balanced and does not increase the maximum average degree. It is shown that for every hypergraph G there exists a balanced extension F of G. Moreover every r-uniform hypergraph has an r-uniform balanced extension. For a graph G let ext (G) denote the minimum number of vertices in any graph that is a balanced extension of G. If G has n vertices, then an upper bound of the form ext (G)<ctn ~ is proved. This is best possible in the sense that ext (G)=..c2n 2 for an infinite family of graphs. However for sufficiently dense graphs an improved upper bound ext (G)<csn can be obtained, confirming a conjecture of P. Erd6s.