Here is a partition of [125] into five parts, each without an arithmetic progression of length 3. This partition witnesses to the fact that W(5,3) > 125.

Block 1:
3 9 14 20 27 29 30 32 39 41 42 47 60 66 71 77 84 86 87 89 96 98 99 104 116 120
Block 2:
5 11 12 15 16 22 43 48 52 54 57 59 67 69 72 73 78 100 105 111 114 115 121
Block 3:
2 4 13 19 23 26 28 35 38 40 56 58 62 64 80 82 83 85 91 92 94 95 113 117 119 123
Block 4:
6 17 21 31 34 44 46 49 50 53 55 63 65 68 70 74 88 97 101 103 107 110 112 122 125
Block 5:
1 7 8 10 18 24 25 33 36 37 45 51 61 75 76 79 81 90 93 102 106 108 109 118 124