For graphs A, B and a positive integer r, the relation means that whenever Δ is an r-colouring of the vertices of A, then there is an embedding ϕ of B into A such that Δ ∘ ϕ is constant. A class of graphs has the Ramsey property if, for every , there is an such that . For a given finite graph G, let Forb(G) denote the class of all finite graphs which do not embed G. It is known that, if G is 2-connected, then Forb() has the Ramsey property, and Forb(G) has the Ramsey property if and only if Forb(G) also has the Ramsey property. In this paper we show that if neither G nor its complement is 2-connected, then either (i) G has a cut point adjacent to every other vertex, or (ii) G has a cut point adjacent to every other vertex except one. We show that Forb(G) has the Ramsey property if G is a path of length 2 or 3, but that Forb(G) does not have the Ramsey property if (i) holds and G is not the path of length 2.

We investigate the spectrum for frames with block size four, and discuss several applications to the construction of other combinatorial designs.

Our main result is that a frame of type hu, having blocks of size four, exists if and only if u ≥ 5, h ≡ 0 mod 3 and h(u — 1) ≡ 0 mod 4, except possibly where

1. (i) h = 9 and u ∈ ﹛13,17,29,33,93,113,133,153,173,193﹜;

2. (ii) h ≡ 0 mod 12 and u ∈ ﹛8,12﹜,

3. h = 36 and u ∈ ﹛7,18,23,28,33,38,43,48﹜,

4. h = 24 or 120 and u ∈ ﹛7﹜,

5. h = 72 and u ∈ 2Z+ U ﹛n : n ≡ 3 mod4 and n ≤527﹜ U ﹛563﹜; or

6. (iii) h ≡ 6mod l2 and u ∈ (﹛17,29,33,563﹜ U ﹛n : n ≡ 3 or 11 mod 12 and n ≤ 527﹜ U ﹛n : n ≡ 7 mod 12 and n ≤ 259﹜), h = 18.

Additionally, we give a new recursive construction for resolvable group-divisible designs from frames: if there is a resolvable k-GDD of type gu, a k-frame of type ﹛mg)v where u ≥ m + 1, and a resolvable TD(k, mv) then there is a resolvable k-GDD of type (mg)uv. We use this to construct some new resolvable GDDs with group size three and block size four.