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In the game SET each card has $4$ properties (color, filling, shape and number), each with $3$ possible values, for a total of $3 ^ 4 = 81$ cards. We can think of cards as elements of $\{0,1,2\} ^ 4$. 3 cards $x,y$ and $z$ are said to form a SET if at every coordinate, they are either all equal or all different. That is, for every $i$, either $x_i = y_i = z_i $ or $\{x_i,y_i,z_i\} = \{0,1,2\}$. More generally, we call any such game with the cards $\{0,1,2\} ^ i$ a SET-like game.

It is easy to see that in a SET-like game, for any $2$ distinct cards, there exists exactly $1$ card that completes them into a set. I want to know if SET-like games are the only structures satisfying this property.

More precisely, let $X$ be a nonempty set and $S \subseteq \{s \subseteq X \mid |s| = 3 \}$ such that for every distinct $x,y\in X$, there exists a unique $z \in X$ so that $\{ x, y, z \} \in S$. Does it follow that $(X, S)$ is isomorphic to some SET-like game?

All I've managed to prove to far is that $|X|$ is odd: given $a \in X$, we can partition $X \setminus \{ a \}$ into pairs, where $ \{ b,c \} $ are a pair if $\{ a, b, c \} \in S$. The main direction to a solution I have in mind is to try to prove that $|X|$ is a power of $3$.

The problem is also equivalent to the following in the language of hypergraphs: what are the possible $3$-uniform, $\frac{|X| - 1}{2}$-regular, linear hypergraphs, $(X, E)$, where $|X|$ is odd?

Ynir Paz
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2 Answers2

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A collection of triples with the property that for any $x,y$ there exists a unique $z$ such that the triplet $\{x,y,z\}$ appears is known as a Steiner triple system (STS). Not all STS’s are SET-like. For example, the Fano plane is an STS supported on a seven element set, consisting of these triplets: $$ (1,2,3), (1,4,5),(1,6,7), (2,4,6),(2,5,7),(3,4,7),(3,5,6) $$ It is known that an STS exists on a set of cardinality $n$ if and only if the remainder of $n\pmod 6$ is either $1$ or $3$, so there are plenty of STS’s which are not SET-like.

Mike Earnest
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Set itself can be recast as affine geometry over $\mathbb{F}_3$: the condition for three cards $\{ x, y, z \} \subset \mathbb{F}_3^4$ to form a set is equivalent to the condition that they all lie on an affine line, and also turns out to be equivalent to the condition that $x + y + z = 0$. So the reason any two cards in a set $\{ x, y, z \}$ uniquely determine the third is that

  • any two points on an affine line determine the line, and
  • over $\mathbb{F}_3$ there are exactly $3$ points on every affine line.

This is explained in more detail in several places, and it also suggests where to look for counterexamples: we can consider projective geometry over $\mathbb{F}_2$. In a projective space over $\mathbb{F}_2$ points can be organized into projective lines, it is still true that two points determine a line, and the projective line over $\mathbb{F}_2$ again has $3$ points. The smallest nontrivial projective space over $\mathbb{F}_2$ is the Fano plane $\mathbb{P}^2(\mathbb{F}_2)$, as Mike says, and it has $2^2 + 2 + 1 = 7$ points so cannot also be an affine geometry over $\mathbb{F}_3$. Similarly projective space $\mathbb{P}^3(\mathbb{F}_2)$ over $\mathbb{F}_2$ has $2^3 + 2^2 + 2 + 1 = 15$ points.

For more you can consult the Wikipedia article on Steiner systems, also as Mike says.

Qiaochu Yuan
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