Let Δ ⊆ [ ω ] 2 \Delta \subseteq {[\omega ]^2} be an undirected graph on ω \omega , and let u ∈ [ 0 , 1 ] u \in [0,\,1] . Following P. Erdös and A. Hajnal, we write ( ω , 2 , u ) ⇒ Δ (\omega ,\,2,\,u) \Rightarrow \Delta to mean: whenever E 1 ⊆ [ 0 , 1 ] {E_1} \subseteq [0,\,1] is a measurable set of Lebesgue measure at least u u for every I ∈ [ ω ] 2 I \in {[\omega ]^2} , then there is some t ∈ [ 0 , 1 ] t \in [0,\,1] such that Δ \Delta appears in the graph Γ t = { I : t ∈ E I } {\Gamma _t} = \{ I:\,t \in {E_I}\} in the sense that there is a strictly increasing function f : ω → ω f:\,\omega \to \omega such that { f ( i ) , f ( j ) } ∈ Γ t \{ f(i),\,f(j)\} \in {\Gamma _t} whenever { i , j } ∈ Δ \{ i,\,j\} \in \Delta . We give an algorithm for determining when ( ω , 2 , u ) ⇒ Δ (\omega ,\,2,\,u) \Rightarrow \Delta for finite Δ \Delta , and we show that for infinite Δ , ( ω , 2 , u ) ⇒ Δ \Delta ,\,(\omega ,\,2,\,u) \Rightarrow \Delta if there is a υ > u \upsilon > u such that ( ω , 2 , υ ) ⇒ Δ ′ (\omega ,\,2,\,\upsilon ) \Rightarrow {\Delta ^\prime } for every finite Δ ′ ⊆ Δ \Delta ’ \subseteq \Delta . Our results depend on a new condition, expressed in terms of measures on β ω \beta \omega , sufficient to imply that Δ \Delta appears in Γ \Gamma (Theorem 2F), and enable us to identify the extreme points of some convex sets of measures (Theorem 5H).