t.nb

matches[wins_, tickets_] := Length[DeleteCases[Map[match[tickets, #] &, wins], {}]] == Length[wins];

match[tickets_, win_] := Select[tickets, Intersection[#, win] == win &, 1];

f[n_, k_, p_] := (
  field = Range[n];

  If[n == k, Return[{field}]];

  wins = Subsets[field, {k}];

  If[k == p, Return[wins]];  

  tickets = Join[wins, Subsets[field, {k + 1, p}]];

  tc = Select[Subsets[tickets], Union[Flatten[#]] == field &];

  pw = Select[tc, matches[wins, #] &];

  Return[First[SortBy[pw, Length]]];
);

f[4,2,3]
(* {{1, 2}, {1, 3, 4}, {2, 3, 4}} *)

f[5,2,3]
(* {{1, 2, 3}, {1, 2, 4}, {1, 2, 5}, {3, 4, 5}} *)