ClearAll[S,Tw,Th,B,m,n0];
B=Binomial;
n0=10;
m=4;
S=Range[n0];
Tw=Subsets[S,{2}];Th=Subsets[S,{m}];

(2^(Length[Tw]))
Plus@@ DigitCount[%]

ClearAll[X,sel,tbl,res];
j=0;
While[j<1000,
sel=RandomChoice[{0,1},Length[Tw]];
X=Pick[Tw,sel,1];
tbl=Table[{k,Intersection[Subsets[Th[[k]],{2}],X]},{k,1,B[Length[S],m]}];
res=Map[If[MatchQ[Length[#[[2]]],0|B[m,2]],{Th[[#[[1]]]]},{}]&,tbl]~Flatten~1;
If[Length[res]<1,Print["j="<>ToString[j]];Print[X];Print[res];Break[]];
j++]//AbsoluteTiming

X

h=Graph[Map[#[[1]]\[UndirectedEdge]#[[2]]&,X]];
HighlightGraph[CompleteGraph[n0,VertexLabels->"Name"],h,GraphHighlightStyle->"Thick"]