In the following list {{1,3,5},{2,4,5},{3,4},{1,5}}, two elements have an empty intersection. This is not the case of the list {{1,4,5},{3,4,5},{3,4},{1,3}}. Is there a simple way to perform such a test? Thanks!

One could compute all the intersections using DistanceMatrix, and test for the presence of an empty list

hasEmptyIntersectionQ[list_] := MemberQ[DistanceMatrix[list, 

   DistanceFunction -> Intersection], {}, {2}] 



hasEmptyIntersectionQ@{{1, 4, 5}, {3, 4, 5}, {3, 4}, {1, 3}}

(* False *)



hasEmptyIntersectionQ@{{1, 3, 5}, {2, 4, 5}, {3, 4}, {1, 5}}

(* True *)

f[list_List] := Not[And @@ Flatten[Outer[IntersectingQ, #, #, 1] &[list]]]



f[{{1, 3, 5}, {2, 4, 5}, {3, 4}, {1, 5}}]

f[{{1, 4, 5}, {3, 4, 5}, {3, 4}, {1, 3}}]

True

False

Edit

Because I like SparseArrays a lot, here a variant involving SparseArray that solves this problem for lists of positive integers quite quickly:

h[list_List] := Module[{A},

  A = SparseArray[

    Join @@ MapIndexed[{x, i} [Function] Thread[{x, i[[1]]}], list] -> 1

    ];

  (A[Transpose].A)["Density"] < 1.

  ]

A timing comparison:

a = Table[RandomInteger[{1, 10}, RandomInteger[{1, 10}]], {i, 1, 1000}];

r1 = f[a]; // AbsoluteTiming // First

r2 = hasEmptyIntersectionQ[a]; // AbsoluteTiming // First

r3 = h[a]; // AbsoluteTiming // First

r1 == r2 == r3

6.89991

2.73299

0.015683

True

Two additional ways, the first short and slow, the second ugly but fast:

ClearAll[hasDisjointPairQa, hasDisjointPairQb]

hasDisjointPairQa = Or @@ DisjointQ @@@ Subsets[#, {2}] &;





hasDisjointPairQb = Module[{a = False}, 

  Do[If[DisjointQ[#[[i]], #[[j]]], a = True; Break], 

  {i, Length[#] - 1}, {j, i + 1, Length@#}]; a] &;



lists1 = {{1, 3, 5}, {2, 4, 5}, {3, 4}, {1, 5}};

lists2 = {{1, 4, 5}, {3, 4, 5}, {3, 4}, {1, 3}};

hasDisjointPairQ /@ {lists1, lists2}

{True, False}

hasDisjointPairQb /@ {lists1, lists2}

{True, False}

Timings using Henrik's setup (f and h are from Henrik's answer and hasEmptyIntersectionQ from Jason B.'s):

SeedRandom[1]

a = Table[RandomInteger[{1, 10}, RandomInteger[{1, 10}]], {i, 1, 1000}];



f[a] // AbsoluteTiming

{8.23578, True}

hasDisjointPairQa[a] // AbsoluteTiming

{4.10347, True}

hasEmptyIntersectionQ[a] // AbsoluteTiming

{3.08455, True}

h[a] // AbsoluteTiming

{0.0256391, True}

hasDisjointPairQb[a] // AbsoluteTiming

{0.000284839, True}

Frankly, I'm not quite clear about your question.
If you want the testing function to return True only when any two elements of the list have nonempty intersection, Jason B. and Henrik Schumacher gave the correct answers. If you want the testing function to return True as far as there are two elements of the list have nonempty intersection, the following function would work:

f[x_]:=DuplicateFreeQ[x//Flatten]