1. Every H-decomposition of K n has a nearly resolvable alternative Wilson: e(H) | n(n-1)/2 and gcd(H) | n-1 n>> then there exists an H-decomposition of K n. There may be many distinct H-decompositions of K n which vary in their properties. Example: H=K k k | n k-1 | n-1 n>> then there exists a resolvable K k dec. of K n. This is a theorem of Ray-Chaudhuri & Wilson. There are also non-resolvable ones. There is no analog of the theorem of RC & W for general graphs H. In fact it is not true for some H. (e.g. H=K 1,t where t > 2 is odd). The resolution number  (H,n): let L be an H-dec. of K n.  (L) is the chromatic number of the intersection graph of L.  (H,n) = min L  (L).

2. By RC & W  (K k,n) = (n-1)/(k-1) iff n=k mod k(k-1) and n >>. Trivially,  (H,n)  (n-1)h/(2m) where m=|e(H)|. Equality holds iff there is a resolvable H-decomposition. The main result Let H be a fixed graph with h vertices and m edges. Then:  (H,n)  (n-1)h/(2m). The o(n) term is, in fact, of the form n b where b < 1. The error term cannot be omitted. Outline of proof First, we show that if K n is H-dec. then it can also be decomposed into H-decomposable cliques whose sizes are bounded. – This follows from a theorem of Wilson regarding pairwise balanced designs, together with an additional simple set-theoretic argument. We also need to use the powerful theorem of Pippenger & Spencer regarding the chromatic index of uniform hypergraphs:

3. Let h and C be positive integers and let  and   be positive reals. There exists N 0 =N 0 (h,C,  ) and 0 N 0 vertices and: –There exists d >  n such that for every vertex x |deg(x)-d| < d   –Any two vertices appear together in at most C edges. Then, q(S) < d+d   Every H-dec. defines an h-uniform hypergraph whose edges correspond to the vertices of each member of the decomposition. Clearly, the chromatic index of this hypergraph is what we need to bound. We need to show there is an H-dec. whose hypergraph satisfies the conditions of P & S with d=(n-1)h/(2m). We need the following large deviation result: –For every  0 there exists t=t(  ) such that if t >T and X 1,..., X t are t mutually independent discrete r.v. taking values between 0 and  and  is the expectation of X= X 1 +... + X t then: prob[|X-  t 0.51 ] < t -2. Proof is a simple use of Azuma’s inequality.

4. Combining it all together Let F=F(H) be a finite set of integers with the property that if K n is H-dec. then K n is also decomposable into H-decomposable cliques whose sizes belong to F. Define C=(k-1)/  (H) where k is the largest integer in F. Define  =h/(3m) < 1. Let  =0.6, and let  and N 0 be as in P & S. For each f  F let L f be a fixed H-dec. of K f. let Y f be the r.v. corresponding to the number of members of K f containing a randomly selected vertex. Note that Y f is discrete and 0 < Y f < k. We show that if n >> and K n is H- decomposable then  (H,n) < d+d  where d=(n-1)h/(2m). Let L* be a decomposition of K n into H-decomposable cliques whose sizes belong to F. Each Q  L* is isomorphic to some K f so there are f! different ways to decompose Q into copies of H using L f. for each Q  L* we randomly and uniformly choose such a permutation. All |L*| choices are independent. This defines a random H-dec. of K n denoted L.

5. We show that with positive probability, each vertex of K n appears in at least d-d  and in at most d+d  members of L. This follows (with a little work) from the large deviation lemma. Any two members of K n appear together in at most C members of L. The last two claims show that P & S holds for the hypergraph corresponding to L, with positive probability. A conjecture The o(n) error term in the result can be replaced by a constant (which depends only on H). Namely:  (H,n)  (n-1)h/(2m) +C(H). Another small goodie For every H there are infinitely many n, for which there exists an H-dec. L of K n such that the intersection graph of L is regular of degree (n-1)h/(2m). (This is only interesting if H is not a regular graph).