Combinatorics Seminar
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1 SET R and AG(4, 3) Combinatorics Seminar Robert Won (Joint with M. Follett, K. Kalail, E. McMahon, C. Pelland) Partitions of AG(4, 3) into maximal caps, Discrete Mathematics (2014) October 19, / 41
2 The card game SET R SET R is played with 81 cards. Each card is characterized by 4 attributes: Number: 1, 2 or 3 symbols. Color: Red, purple or green. Shading: Empty, striped or solid. Shape: Ovals, diamonds or squiggles. A set is three cards where each attribute is independently either all the same or all different October 19, 2016 An introduction to SET R 2 / 41
3 The card game SET R The number of attributes that are the same can vary. Shape and shading are the same, color and number are different. All attributes are different. October 19, 2016 An introduction to SET R 3 / 41
4 The card game SET R To start the game, twelve SET R cards are dealt face up. If a player finds a set, he takes 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. Image adapted from Davis and Maclagan The Card Game SET October 19, 2016 An introduction to SET R 4 / 41
5 The card game SET R Can you find a set? October 19, 2016 An introduction to SET R 5 / 41
6 Finite affine geometry For affine geometry on a plane, there are three axioms: 1 There exist (at least) 3 non-collinear points. 2 Any two points determine a unique line. 3 Given a line l and a point P not on l, there is a unique line through P parallel to l. The order of a finite geometry is the number of points on each line. Using these axioms, we can draw AG(2, 3), the affine plane of order 3. October 19, 2016 Finite affine geometry 6 / 41
7 Finite affine geometry October 19, 2016 Finite affine geometry 7 / 41
8 Seeing geometry in SET R A deck of SET R cards is a finite affine geometry. The cards are the points; three points are on a line if those three cards form a set. This works because any two cards uniquely determine a third card that completes the set. October 19, 2016 Connecting SET R to geometry 8 / 41
9 Seeing geometry in SET R A deck of SET R cards is a finite affine geometry. The cards are the points; three points are on a line if those three cards form a set. This works because any two cards uniquely determine a third card that completes the set. October 19, 2016 Connecting SET R to geometry 8 / 41
10 Parallel sets We can also see parallel lines as parallel sets. If the original set has any attribute all the same, the parallel set will also have the same attribute all the same. If any attributes are different in the set, you can lay the cards of the parallel set down so that each of those attributes cycle in the same way as in the original. October 19, 2016 Connecting SET R to geometry 9 / 41
11 Finite affine planes of SET R Cards October 19, 2016 Connecting SET R to geometry 10 / 41
12 Finite affine planes of SET R Cards October 19, 2016 Connecting SET R to geometry 10 / 41
13 Finite affine planes of SET R Cards October 19, 2016 Connecting SET R to geometry 10 / 41
14 Finite affine planes of SET R Cards October 19, 2016 Connecting SET R to geometry 10 / 41
15 Finite affine planes of SET R Cards This picture is sometimes called a magic square. Find all the sets in it! October 19, 2016 Connecting SET R to geometry 11 / 41
16 Finite affine planes of SET R Cards This picture is sometimes called a magic square. Find all the sets in it! October 19, 2016 Connecting SET R to geometry 11 / 41
17 A finite affine hyperplane Select any remaining card and construct two more magic squares. This creates a hyperplane. October 19, 2016 Connecting SET R to geometry 12 / 41
18 A finite affine hyperplane epresented by three side-by-side 3 3 grids. Again, three points are colline inta(0, 0, 0). In Figure line in AG(3, 3): 2, three collinear points are shown.!!! Figure 2. AG(3, 3) with one set of collinear points shown. epresented by a 9 9 grid, consisting of nine 3 3 grids. A line will be th n the same subgrid, or in three subgrids that correspond to a line in AG(2, 3). in AG(4, 3) whose anchor point is in the upper left. You can verify that the c oints, where the third point completing the line for each pair is the point in October 19, 2016 Connecting SETr to geometry 13 / 41
19 The entire deck (AG(4, 3)) October 19, 2016 Connecting SET R to geometry 14 / 41
20 5-Attribute SET R October 19, 2016 Connecting SET R to geometry 15 / 41
21 Some easy counting How many cards (points) are there? October 19, 2016 Connecting SET R to geometry 16 / 41
22 Some easy counting How many cards (points) are there? 81 = 3 4 October 19, 2016 Connecting SET R to geometry 16 / 41
23 Some easy counting How many cards (points) are there? How many sets (lines) are there? 81 = 3 4 October 19, 2016 Connecting SET R to geometry 16 / 41
24 Some easy counting How many cards (points) are there? How many sets (lines) are there? 81 = = (81 80)/3! = ( 81 2 ) /3 October 19, 2016 Connecting SET R to geometry 16 / 41
25 Some easy counting How many cards (points) are there? How many sets (lines) are there? 81 = = (81 80)/3! = ( 81 2 ) /3 How many sets through a given card are there? October 19, 2016 Connecting SET R to geometry 16 / 41
26 Some easy counting How many cards (points) are there? How many sets (lines) are there? 81 = = (81 80)/3! = ( 81 2 ) /3 How many sets through a given card are there? 40 = 80/2 October 19, 2016 Connecting SET R to geometry 16 / 41
27 SET R and F n 3 We can also think of a deck of SET R cards as the vector space F 4 3. Each attribute corresponds to a coordinate, which can take on one of three possible values Number Color Shading Shape 1 1 red 1 empty 1 oval green 2 striped 2 diamond purple 0 solid 0 squiggle 0 October 19, 2016 Caps and partitions in F n 3 17 / 41
28 SET R and F n 3 With this choice of coordinates: The first set has coordinates (1, 0, 2, 1), (2, 1, 2, 1) and (0, 2, 2, 1). The second set has coordinates (2, 1, 0, 0), (1, 2, 1, 1), (0, 0, 2, 2). Three cards form a set if and only if the vector sum is 0 mod 3 they are of the form x, x + a, x + 2 a for some a 0 October 19, 2016 Caps and partitions in F n 3 18 / 41
29 SET R and F n 3 In F n 3, we define a line (a.k.a. an algebraic line) to be three points that sum to 0 mod 3 three points of the form x, x + a, x + 2 a for some a 0 Now we have linear (well, affine) algebra! The maps F n 3 Fn 3 taking lines to lines are precisely the affine transformations for A GL(n, 3) x A x + b October 19, 2016 Caps and partitions in F n 3 19 / 41
30 Caps A k-cap is a collection of k points with no three collinear. 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 F 2 3, maximal caps contain four points October 19, 2016 Caps and partitions in F n 3 20 / 41
31 Caps A k-cap is a collection of k points with no three collinear. 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 F 2 3, maximal caps contain four points October 19, 2016 Caps and partitions in F n 3 20 / 41
32 Caps A complete 8-cap: A complete (and maximal) 9-cap: October 19, 2016 Caps and partitions in F n 3 21 / 41
33 Caps Two caps c 1, c 2 are called equivalent if there exists an affine transformation mapping c 1 to c 2. Fact: All maximal caps in F n 3 are equivalent for n 6. n = 4, the Pellegrino cap (Hill 1983) n = 5, the Hill cap (Edel, Ferret, Landjev, Storme 2002) n = 6, (Potechin 2008) Open question: n > 6? October 19, 2016 Caps and partitions in F n 3 22 / 41
34 An integer sequence Denote by r 3 (F n 3 ) the size of a maximal cap in Fn 3 October 19, 2016 Caps and partitions in F n 3 23 / 41
35 An integer sequence Terry Tao s blog: Open question: best bounds for cap sets ( Perhaps my favourite open question is the problem on the maximal size of a cap set a subset of F n 3 which contains no lines... n = r 3 (F n 3 ) r 3 (F n 3 ) October 19, 2016 Caps and partitions in F n 3 24 / 41
36 An integer sequence (Almost) trivially, 2 n r 3 (F n 3 ) < 3n. Asymptotically... Edel (2004): r 3 (F n 3 ) = Ω(2.2174n ). Brown-Buhler (1982, JCTA): r 3 (F n 3 ) = o(3n ). Meshulam (1995, JCTA): r 3 (F n 3 ) = O(3n /n). Bateman-Katz (2012, J AMS): r 3 (F n 3 ) = O(3n /n 1+ɛ ) Ellenberg-Gijswijt (to appear in Annals) On large subsets of F n q with no three-term arithmetic progression : r 3 (F n 3 ) = O(2.756n ) (!!!) Ellenberg-Gijswijt adapt a method of Croot-Lev-Pach (to appear in Annals). The punchline: asymptotically, n < r 3 (F n 3 ) < 2.756n October 19, 2016 Caps and partitions in F n 3 25 / 41
37 Maximal caps in AG(4, 3) Remember this cap? Consider this card. October 19, 2016 Caps and partitions in F n 3 26 / 41
38 Maximal caps in AG(4, 3) Theorem (F, K, M, P, -, 2014 ) (First observed by Gary Gordon) Every 20-cap in AG(4, 3) consists of ten lines intersecting at one point with the point of intersection removed. We call this point the anchor point. October 19, 2016 Caps and partitions in F n 3 27 / 41
39 Maximal caps in AG(4, 3) Theorem (F, K, M, P, -, 2014 ) Any two maximal caps in AG(4, 3) with different anchor points intersect 0 37 October 19, 2016 Caps and partitions in F n 3 28 / 41
40 Partitioning AG(4, 3) (Tony Forbes) AG(4, 3) can be partitioned into 4 disjoint 20-caps and their anchor point. October 19, 2016 Caps and partitions in F n 3 29 / 41
41 Partitioning AG(4, 3) October 19, 2016 Caps and partitions in F n 3 30 / 41
42 Partitioning AG(4, 3) Are all partitions of AG(4, 3) equivalent? October 19, 2016 Caps and partitions in F n 3 31 / 41
43 Linear transformations When the anchor point is fixed at 0, affine transformations are linear transformations. Here s one: October 19, 2016 Equivalence classes of partitions 32 / 41
44 Spot the difference Consider these two caps with respect to our favorite cap, S. October 19, 2016 Equivalence classes of partitions 33 / 41
45 Spot the difference The first is 1-completable. The second is 2-completable. October 19, 2016 Equivalence classes of partitions 34 / 41
46 Spot the difference 6-completables, too! October 19, 2016 Equivalence classes of partitions 35 / 41
47 Completability 198 caps disjoint from S. With respect to our favorite cap, S: 36 1-completable caps 90 2-completable caps 72 6-completable caps Every partition consists of: S, 1-completable, 6-completable, 6-completable S, 2-completable, 6-completable, 6-completable October 19, 2016 Equivalence classes of partitions 36 / 41
48 Linear transformations Theorem (F, M, K, P, -, 2014) Let T be an affine transformation fixing S: T(n-comp) is an n-comp, n {1, 2, 6} No affine transformations exist between 1-completables and 2-completables. October 19, 2016 Equivalence classes of partitions 37 / 41
49 Partition classes 216 different partitions of AG(4, 3) with S = 216 Two equivalence classes (no affine transformations): E 1 : 36 partitions {S, 1-comp, 6-comp, 6-comp} E 2 : 180 partitions {S, 2-comp, 6-comp, 6-comp} Each 6-completable once in E 1 and five times in E 2. October 19, 2016 Equivalence classes of partitions 38 / 41
50 Linear transformations of E 2 Suppose D 2 E 2, and let S 2 be the 2-completable of D 2. 8 linear transformations fix D 2 cap-wise ( = Z 4 Z 2 ). 8 linear transformations fix S and S 2 and switch 6-completables. Thus, a group of order 16 fixes D 2 set-wise ( = Z 4 Z 4 ). Another set of 16 linear transformations fix S and S 2 but send 6-completables to two new 6-completables. Thus, group of order 32 fixing S and S 2 ( = (Z 8 Z 2 ) Z 2 ). October 19, 2016 Equivalence classes of partitions 39 / 41
51 Linear transformations of E 1 Suppose D 1 E 1, and let S 1 be the 1-completable of D transformations fix D 1 cap-wise ( = Z 4 D 5 ). Also, 40 transformations fix S and S 1 and switch 6-completables. Thus, group of order 80 fixing S and S 1. Isomorphic to Z 20 Z 4. October 19, 2016 Equivalence classes of partitions 40 / 41
52 Summary Every maximal cap in AG(4, 3) consists of ten lines intersecting at an anchor point. AG(4, 3) can be partitioned into four disjoint maximal caps and their anchor point. There are two equivalence classes of partitions. Also interesting Building complete caps (Jordan Awan): October 19, 2016 Equivalence classes of partitions 41 / 41
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