Here is a partition of [194] into 6 parts, each without an arithmetic progression of length 3. This partition witnesses to the fact that W(6,3) > 194.

Block 1:
1 4 8 20 24 26 35 37 43 53 55 56 58 64 68 74 76 89 121 130 131 134 136 145 146 149 150 157 172 175 176 182 185 190 191
Block 2:
7 9 13 21 31 34 39 45 52 63 67 75 78 79 85 94 99 101 102 112 114 115 133 138 142 144 148 156 160 166 180 192
Block 3:
11 18 36 47 50 60 62 65 69 71 81 86 87 90 96 98 108 116 117 128 137 141 143 152 155 164 168 171 179 183 189
Block 4:
3 16 22 25 27 33 40 46 48 49 54 57 72 91 103 106 118 120 124 126 127 129 135 139 153 159 162 169 170 174 193
Block 5:
6 10 12 15 17 23 30 32 42 44 51 59 66 77 80 84 93 104 105 107 111 123 125 132 140 147 161 165 177 181 186 188 194
Block 6:
2 5 14 19 28 29 38 41 61 70 73 82 83 88 92 95 97 100 109 110 113 119 122 151 154 158 163 167 173 178 184 187