The card game SET, finite affine geometry, andcombinatorial number theory

Portland State UniversityColloquium

Robert WonUniversity of Washington

May 24, 2019 1 / 33

The card game SET• SET is played with deck of 81 cards. On each card, there are

symbols with four attributes:

Number: 1 2 3Color: red purple green

Pattern: empty striped solidShape: ovals diamonds squiggles

• A SET is a collection three cards such that, for each attribute, thecards are either all the same or all different.

May 24, 2019 An introduction to SET 2 / 33

The card game SET

The number of attributes that are the same can vary.

Shape and pattern are the same,color and number are different.

All attributes are different.

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The rules of SET

• To start the game, twelve SET cards are dealt face up.• All players look for SETs simultaneously.• If a player finds a SET, they take it and three new cards are dealt.• If there are no SETs, three more cards are dealt. The three cards are

not replaced on the next SET, reducing the number back to twelve.• The player who finds the most SETs is the winner.

SET is a popular game among mathematicians (and amongnon-mathematicians). So popular that...

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May 24, 2019 An introduction to SET 5 / 33

The rules of SET• To start the game, twelve SET cards are dealt face up.• All players look for SETs simultaneously.• If a player finds a SET, they take it and three new cards are dealt.• If there are no SETs, three more cards are dealt.• The player who finds the most SETs is the winner.

QuestionHow many cards are needed to guarantee existence of a SET?

Question (Rephrased)How big is the largest collection of cards containing no SETs?

Question (More specific)What is the largest collection of cards containing no SETs?

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Finite affine geometry

• A deck of SET cards forms a finite geometry.• Key observation: Any two SET cards determine a unique SET:

• Geometry axiom: Any two points determine a unique line.• So the cards are the points and the SETs are the lines.

May 24, 2019 Finite affine geometry 7 / 33

Finite affine geometry

• Any two points determine a unique line. Any three non-collinearpoints determine a unique plane. What is a plane of SET cards?

• How many lines does this plane contain? (How many SETs?)

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Finite affine geometry

There are 12 SETs: Each pair of points in the plane determines a line inthe plane.

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A 3-dimensional flat

Select any remaining card. Along with the three initial cards, thesefour cards should determine a 3-dimensional flat.

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A 3-dimensional flat

A line in this 3-dimensional flat.AG(3, 3) is represented by three side-by-side 3 × 3 grids. Again, three points are collinear if their sum

mod 3 is the point (0, 0, 0). In Figure 2, three collinear points are shown.

! !!

Figure 2. AG(3, 3) with one set of collinear points shown.

AG(4, 3) is represented by a 9 × 9 grid, consisting of nine 3 × 3 grids. A line will be three points thatappear either in the same subgrid, or in three subgrids that correspond to a line in AG(2, 3). Figure 3 showsa complete cap in AG(4, 3) whose anchor point is in the upper left. You can verify that the cap does consistof 10 pairs of points, where the third point completing the line for each pair is the point in the upper left.

Figure 3. A complete cap in AG(4, 3) whose anchor point is in the upper left.

This grid can also be realized as a way of viewing points with 4 coordinates, where the first two coordinatesgive the subgrid, and the second two give the point within that subgrid. In this case, the cap S picturedabove is the first complete cap lexicographically.

Let S1 be an arbitrary cap with anchor point a. Since the coordinates of a line sum to (0, 0, 0, 0), if thecoordinates for the points in S1 are summed with 10a, the result must be (0, 0, 0, 0), since this is sum of 10lines. Thus, the sum of the points in S1 is −a mod 3.

A computer search shows that any two caps with different anchor points necessarily intersect. This factis interesting; it would be useful to have a geometric proof.

Another computer search verifies that there are 8424 complete caps with anchor a. These are all affinelyequivalent, and the affine transformation that take one to another is actually a linear transformation, sincethe anchor point (0, 0, 0, 0) must go to itself. This reduces the transformations we must consider to GL(4, 3).

It is possible to decompose all of AG(4, 3) into 4 mutually disjoint complete caps with anchor a. Onesuch decomposition, where one of the complete caps is S, pictured above, is shown in Figure 4

Figure 4. A complete cap in AG(4, 3) whose anchor point is in the upper left.

2

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A 4-dimensional flatEach card has four attributes: the deck is a 4-dimensional geometry.

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A 5-dimensional flat

And we’re done! But... we’re mathematicians.

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SET and Fn3

We can identify SET cards with points of F43 = (Z/3Z)4:

Number: 1 2 3Color: red purple green

Pattern: empty striped solidShape: ovals diamonds squiggles

1 2 0

(2, 2, 1, 1) (2, 0, 2, 0) (2, 1, 0, 2)

Key observation: Three cards form a SET if and only if they sum to 0.

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SET, geometry, and arithmetic

Question (Rephrased)How big is the largest collection of cards containing no SETs?

Question (Geometric version)

How big is the largest subset of F43 such that no three of the

points are collinear?

Question (Arithmetic version)

How big is the largest subset of F43 containing no three distinct

x,y, z such that x + y + z = 0?

Question (Arithmetic version)’

How big is the largest subset of F43 containing no x, x + a, x + 2a,

i.e., no three-term arithmetic progressions?

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SET and combinatorial number theory

Question (Combinatorial number theory version)

How big is the largest subset of F43 containing no x, x + a, x + 2a,

i.e., no three-term arithmetic progressions?

Analogous questionHow big is the largest subset of 1, 2, . . . ,N (or Z/NZ)containing no three-term arithmetic progressions?

A classic question in combinatorial number theory.

Theorem [Roth (1953)]Asymptotically, o(N). (Grows less than linearly in N).

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Caps• A capset or cap in Fn

3 is a collection of points with no threecollinear (containing no three-term arithmetic progressions).• A complete cap is a cap for which any other point in the space

makes a line with a subset of points from the complete cap.• A maximal cap is a cap of maximum size.• In F2

3, all complete caps are maximal and contain four points.

w w ww w ww w w

(0, 0), (0, 1), (1, 0), (1, 1)

May 24, 2019 Caps in Fn3 17 / 33

Caps

Generalize this construction to a complete cap in F33 of size 8:v v t

t t ttv vtv v tt t ttv vt

t t tt t tt t tBut this isn’t maximal! A (maximal) cap of size 9:v vtt t tv vt

t t tt tvt t tt tvv vtt tv

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Maximal caps

Denote by r(Fn3) the size of a maximal cap in Fn

3 .• r(F1

3) = 2• r(F2

3) = 4• r(F3

3) = 9• r(F4

3) = 20 [Pellegrino (1971)]. This is the answer for SET.• r(F5

3) = 45 [Edel, Ferret, Landjev, and Storme (2002)]• r(F6

3) = 112 [Potechin (2008)]

Open questionWhen n ≥ 7, what is r(Fn

3)?

This is an interesting integer sequence...

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An integer sequence

... let’s check OEIS!

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An integer sequence

• Terry Tao’s blog: “Open question: best bounds for cap sets”( http://terrytao.wordpress.com/2007/02/23/open-question-best-bounds-for-cap-sets/)

• “Perhaps my favourite open question is the problem on themaximal size of a cap set—a subset of Fn

3 which contains nolines...”

n = 1 2 3 4 5 6 7 8 9 10r(Fn

3) ≥ 2 4 9 20 45 112 236 496 1064 2240r(Fn

3) ≤ 2 4 9 20 45 112 291 771 2070 5619

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Asymptotics of r(Fn3)

(Almost) trivially, 2n ≤ r(Fn3) < 3n.

Asymptotically...• Edel (2004): r(Fn

3) = Ω(2.2174n).• Brown and Buhler (1982, JCTA): r(Fn

3) = o(3n).• Meshulam (1995, JCTA): r(Fn

3) = O(3n/n).• Bateman and Katz (2012, JAMS): r(Fn

3) = O(3n/n1+ε)

• Ellenberg and Gijswijt (2017, Annals): r(Fn3) = O(2.756n) (!!!)

• Adapt a method of Croot, Lev, and Pach (2017, Annals).• CLP: 5/5/16 E: 5/12 G: 5/14 EG: 5/30

Punchline: The asymptotics of r(Fn3) are well-understood. As n→∞:

2.2174n < r(Fn3) < 2.756n.

May 24, 2019 Caps in Fn3 22 / 33

Generalizations of caps

DefinitionA cap in Fn

3 is a collection of points with no three points on acommon line.

Definition [Huang, Tait, W (2019)]A 2-cap in Fn

3 is a collection of points with no three points on acommon line and no four points on a common plane.

Definition [Huang, Tait, W (2019)]Fix d ≥ 1. A d-cap in Fn

3 is a collection of points such that foreach k = 1, . . . , d, no k + 2 points lie on a commonk-dimensional flat. So an ordinary cap is a 1-cap.

Remark: Bennett calls this a (d + 2)-general set in arXiv:1806.05303.

May 24, 2019 Generalizations of caps 23 / 33

Studying 2-capsLet r(d,Fn

3) denote the size of a maximal d-cap in Fn3 . So

r(Fn3) = r(1,Fn

3).

n = 1 2 3 4 5 6 7 8 9 10r(1,Fn

3) ≥ 2 4 9 20 45 112 236 496 1064 2240r(1,Fn

3) ≤ 2 4 9 20 45 112 291 771 2070 5619

And as n→∞, we have 2.2174n < r(1,Fn3) < 2.756n.

QuestionWhat is r(2,Fn

3) for small n?

QuestionHow does r(2,Fn

3) behave asymptotically as n→∞?

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Studying 2-caps

A maximal 2-cap in F23:

w w ww w ww w w

Of course, r(2,Fn3) ≤ r(1,Fn

3). For fixed n, r(d,Fn3) is a non-increasing

function of d.

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Studying 2-caps

A maximal 2-cap in F33:t t tt t tt t t

t t tt t tt t tt t tt t tt t t

v vv v vCan enumerate all possibilities: r(2,F3

3) = 5.

Computationally feasible for n ≤ 6 (for us), C++ code available online:• r(2,F4

3) = 9• r(2,F5

3) = 13• r(2,F6

3) = 27When n ≥ 7, no longer feasible to enumerate all possibilities.

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Studying 2-caps

• Three points in Fn3 are collinear if and only if they sum to ~0 if and

only if they form a three-term arithmetic progression.• What does it mean (arithmetically) for four points to be coplanar?

a b −a− b

c −a + b + c a− b + c

−a− c a + b− c −b− c

• Observation: Given any four points x,y, z,w, no three of whichare collinear, it is possible to relabel them so that x + y = z + w.

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Sidon sets

DefinitionLet G be an abelian group. A subset A ⊆ G is called a Sidon setif, whenever a + b = c + d for a, b, c, d ∈ A, the pair (a, b) is apermutation of the pair (c, d).

• Generalization of a three-term arithmetic progression-free set,since if x, x + a, x + 2a is an arithmetic progression in G, then

(x + a) + (x + a) = x + (x + 2a).

• Well-studied objects in combinatorial number theory. (SeeO’Bryant’s Dynamic Survey in Electron. J. Comb.)• Sidon to Erdos: What is the largest Sidon set in 1, 2, . . . ,N?

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Sidon sets and 2-caps

Theorem [Huang, Tait, W (2019)]A set of points in Fn

3 is a 2-cap if and only if it is a Sidon set.

Lemma [Cilleruelo, Ruzsa, and Vinuesa (2010)]If C is a Sidon set in an abelian group G, then

|C|(|C| − 1) ≤ |G| − 1.

Corollary [Huang, Tait, W (2019)]

So r(2,Fn3) ≤ d3n/2e.

This seems too easy... but the bound is actually tight!

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Sidon sets and 2-caps

Example (Folklore cf. [Cilleruelo, Ruzsa, Vinuesa (2010)])Let q be an odd prime power and let Fq denote the finite field oforder q. Then the set

(x, x2) | x ∈ Fq

is a Sidon set in Fq × Fq.

• The Sidon property is about the additive structure of Fn3 .

• If n is even, then Fn3∼= Fn/2

3 × Fn/23∼= F3n/2 × F3n/2 .

• The above construction gives a Sidon set of size 3n/2 in Fn3 .

Corollary [Huang, Tait, W (2019)]

When n is even, r(2,Fn3) = 3n/2.

Corollary [Huang, Tait, W (2019)]

Asymptotically, r(2,Fn3) = Θ(3n/2).

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Open questions

Open questionWhen n is odd, what is r(2,Fn

3)? (Or even, when n = 7.)

n = 1 2 3 4 5 6 7 8 9 10r(1,Fn

3) ≥ 2 4 9 20 45 112 236 496 1064 2240r(1,Fn

3) ≤ 2 4 9 20 45 112 291 771 2070 5619

n = 1 2 3 4 5 6 7 8 even oddr(2,Fn

3) ≥ 2 3 5 9 13 27 33 81 3n/2 3(n−1)/2 + 1r(2,Fn

3) ≤ 2 3 5 9 13 27 47 81 3n/2 ⌈3n/2⌉

Bennett adapts the Ellenberg-Gijswijt method to bound the size ofd-caps in Fn

q for all d, q, and n. One case is r(2,Fn3) ≤ 3.821n vs our 3n/2.

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Open questions

Open question

Is the Sidon set (x, x2) the only 2-cap in Fn3?

Open questionWhen d ≥ 3, what is r(d,Fn

3) for small n?

Open questionDoes the arithmetic formulation of a d-cap have anysignificance?

Open questionWhen q 6= 3, what is r(2,Fn

q)?

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Thank you!

And thanks to my coauthors:

Yixuan (Alice) Huang Mike Tait

May 24, 2019 33 / 33