SET® Mathematics
No SET Mapping
The largest group of cards you can put together without creating a set is 20. By following this method, you'll understand how. 
If we choose just two characteristics of a card (for example, shape and number), we can then plot it on a matrix. 
So now we can see if three cards form a set by noticing
if they make a line on the matrix, as shown below. These three cards
all have one object and different shapes.

Similarly, the following lines all produce sets, even if they wrap around the matrix in space like the one on the far right. 
Now the following following matrix shows us how many squares we can fill without creating a line: 4. 
Now we can add a third characteristic, color...and we can think of the matrix as a depiction of a 3D tic tac toe board. You can see below how to plot 9 cards without a set. 
When using all four SET® card properties, we can plot a 20 card no set. 