SET is an interesting card game from a billion years ago which I haven’t thought about in a long time.
There are 81 cards in the deck, and every card is unique. Each card has:
[ul]
[li]A shape (circle, diamond, or squiggle);[/li][li]A color (purple, green, or red);[/li][li]A number (one, two, or three);[/li][li]A shading (solid, checked, or open)[/li][/ul]
A set is a hand of three cards such that for all four properties, the three cards are either all the same, or all different.
Wikipedia states: “A Cap set is a mathematical structure describing a Set layout in which no set may be taken. The largest group of cards you can put together without creating a set is 20.”
The citation for this is a proof, found here(note, this link is a PDF and the proof seems a little… dense.)
The paper states that “any collection of 20 cards must contain a Set” which is slightly different from the Wikipedia entry I quoted, which implies that any collection of 21 cards must contain a Set. Since the Wikipedia site uses the proof as a citation and the proof is hosted at what appears to be the official site for Set, I assume the answer to the OP’s question, “What is the maximum number of cards you can deal onto the table with no set appearing?” is 19.
The number of SET cards out on the table with no sets present before you start to feel like the entire gaming party has gone nuts, stupid, or blind is approximately 15. The difference between this number and the actual possible no-set number may possibly have driven a friend of mine to swear off that particular game for life.