Rota s Conjecture. Jim Geelen, Bert Gerards, and Geoff Whittle

Transcription

1 Rota s Conjecture Jim Geelen, Bert Gerards, and Geoff Whittle

2 Rota s Conjecture For each finite field field F, there are at most a finite number of excluded minors for F-representability.

3 Ingredients of Proof 1. Corollaries of structure theorems for members of proper minor-closed classes of F-representable matroids. 2. Reduction to matroids that are in some sense highly connected. 3. Techniques that generalise those described in Notices paper.

4 Connectivity Functions (A, B) partition of E(M). Then λ M (A) = r(a) + r(b) r(m). If λ M (A) < k, and A, B k, then (A, B) is a k-separation of M. If M is represented, then λ M (A) is the rank of A B.

5 Connectivity Functions (A, B) partition of E(M). Then λ M (A) = r(a) + r(b) r(m). If λ M (A) < k, and A, B k, then (A, B) is a k-separation of M. If M is represented, then λ M (A) is the rank of A B.

6 How can λ behave? Can represent schematically using blob diagrams.

7 Tutte Connectivity M is (Tutte) k-connected if it has no l-separations for l k.

8 Tutte Connectivity M is (Tutte) k-connected if it has no l-separations for l k. Theorem (Notices Paper) There exists a k such that every k-connected excluded minor has at most 2k elements.

9 f -connectivity f an integer-valued function. Then M is f -connected if, whenever (A, B) is a k-separation of M, then either A f (k) or B f (k).

10 f -connectivity f an integer-valued function. Then M is f -connected if, whenever (A, B) is a k-separation of M, then either A f (k) or B f (k). Theorem Let M be an excluded minor for F-representability. Then, for all k there exists n such that, if (A, B) is a k-separation of M, then either A n or B n. This was topic of talk in Princeton 2014.

11 This Talk Outline proof that f -connected excluded minors have bounded size.

12 Note In any unexplained context, F is a finite field; M is an excluded minor for F.

13 Hypotheses The proof uses unwritten-up theorems from matroid minors project. We ll distinguish these by calling such results Hypotheses. In fact for this talk we only need one, but no harm in seeing them all.

14 Hypothesis 1 F-representable matroids are well quasi ordered under the minor order.

15 Big blobs are identified by high-order tangles.

16 Tangle Talk The tangle axioms. A tangle T of order θ + 1 is a collection of subsets of E(M) such that If X T, then λ(x ) θ 1. If (X, Y ) is a θ-separation, then either X or Y is in T. If X, Y, Z T, then X Y Z = E(M). E(M) {e} / T.

17 Tangle Talk The tangle axioms. A tangle T of order θ + 1 is a collection of subsets of E(M) such that If X T, then λ(x ) θ 1. If (X, Y ) is a θ-separation, then either X or Y is in T. If X, Y, Z T, then X Y Z = E(M). E(M) {e} / T. If (X, Y ) is a θ-separation and X T, we will say that X is T -small and that Y contains T.

18 Rank in Tangles Say Z is contained in a T -small set. Then r T (Z), the rank of Z in T is defined by r T (Z) = min{λ(x ) : X Z; X is T -small}.

19 Rank in Tangles Say Z is contained in a T -small set. Then r T (Z), the rank of Z in T is defined by r T (Z) = min{λ(x ) : X Z; X is T -small}. Lemma r T is the rank function of a matroid.

20 Fragile Elements If N is a minor of M, an element e is N-fragile if at least one of M/e or M\e does not have N as a minor.

21 Fragile Elements If N is a minor of M, an element e is N-fragile if at least one of M/e or M\e does not have N as a minor. Hypothesis 2 Let M be an F-representable matroid, T be a tangle in M. Then the set of N-fragile elements has bounded rank in T

22 Relaxation Recall that a relaxation M 2 of M 1 is obtained by turning a circuit-hyperplane into a basis. For all e, either M 1 \e = M 2 \e or M 1 /e = M 2 /e. Relaxation does not behave well wrt representability. But what about wheels and whirls?

23 Hypothesis 3 There exists an integer h such that the following holds. If M 1 and M 2 are F representable matroids on a common ground set, T is a tangle in M, and X is the set of elements such that M 1 \x = M 2 \x for x X. Then r(x ) h. Hypothesis 3 is a lot of work :(

24 Domination Let f and f be functions. Then f dominates f if f (x) f (x) for all x. Note that statement like excluded minors are f -connected remain true for any function f that dominates f.

25 Keeping Connectivity We have a very large excluded minor. We know it is f -connected with f (0) = 0 and f (1) = 1. We would like to remove lots of elements and keep f -connectivity when we remove arbitrary subsets of them. We can t do that, but we can get close.

26 Keeping Connectivity We have a very large excluded minor. We know it is f -connected with f (0) = 0 and f (1) = 1. We would like to remove lots of elements and keep f -connectivity when we remove arbitrary subsets of them. We can t do that, but we can get close.

27 (f, k)-connectivity Just like f -connectivity except that we only require it for {0, 1,..., k}.

28 An induced tangle If M is (f, k)-connected matroid with more than 3f (k) elements, then there is an obvious tangle of order k induced by (f, k)-connectivity. In the same way, f -connectivity induces an obvious maximal tangle.

29 If T is a tangle in M, then X is T -independent if X is independent in M(T ) the matroid of T.

30 Finding connectivity we can keep Le f be a function with f (0) = 0 and f (1) = 2 (for technical reasons). Then f is defined by f (n) = f (n) + n. Note that f (1) = 3 This is problematic.

31 Finding connectivity we can keep Le f be a function with f (0) = 0 and f (1) = 2 (for technical reasons). Then f is defined by f (n) = f (n) + n. Note that f (1) = 3 This is problematic. M is strongly f -connected, if M is f -connected and M has no triangle-triads.

32 Theorem let f be a function with f (0) = 0 and f (1) = 2. Then there exists a function f with f (1) = 0 and f (1) = 2 that dominates f, and a 2-variable function γ such that, if M is an F-representable f -connected matroid, E(M) γ(n, k), and T is the tangle induced by f, then there is a T -independent set X such that either M\X is strongly (f, k)-connected for all X X, or M/X is strongly (f, k)-connected for all X X.

33 We need a theorem of GGHZ. But first we need to know about spikes in matroids.

34 We need a theorem of GGHZ. But first we need to know about spikes in matroids. Theorem There is a function f such that, for all k, there is an n such that, if M is F-representable and (f, k)-connected, E n, λ(x ) k, then either there is an e E X such that M\e or M/e is (f, k)-connected, or there is a spike {L 1,..., L n } such that X L i 1 for all i.

35 Spikes are not so bad If M is (f, k)-connected and {a, b} is a leg of a spike in M, then M\a/b is (f, k)-connected.

36 Lemma Let T be a tangle in M, and C D be a T -independent set such that M/C\D is (f, k)-connected, C D 6n, and T has high enough order, then, up to duality, there is an n-element set X C D such that M\X is strongly ( f, k)-connected for all X X.

37 Lemma Let T be a tangle in M, and C D be a T -independent set such that M/C\D is (f, k)-connected, C D 6n, and T has high enough order, then, up to duality, there is an n-element set X C D such that M\X is strongly ( f, k)-connected for all X X. With these ingredients, the proof of our connectivity result is not difficult.

38 Revision on fixed elements and clones What we really need to know is Lemma If e is fixed in M, then no representation of M\e extends to two projectively inequivalent representations of M.

39 We need to control the number of non-fixed elements in our excluded minor. Fortunately we have the next cool result of GZ. Theorem If T is a tangle in a 3-connected F-representable matroid, then the set of non-fixed elements has bounded rank in T.

40 Weakly fixed elements We cannot quite use the previous theorem as in our strongly f -connected matroids we may have arbitrarily many series classes. But the issue can be dealt with by the notion of weakly fixed elements.

41 Weakly fixed elements We cannot quite use the previous theorem as in our strongly f -connected matroids we may have arbitrarily many series classes. But the issue can be dealt with by the notion of weakly fixed elements. Corollary If k is large enough and M is a strongly ( f, k)-connected, F-representable matroid, then M has a bounded number of elements that are not weakly fixed.

42 Inequivalent Representations Not surprisingly, these are an issue. But f -connectivity is very strong and we can use another nice theorem of GGHZ.

43 Inequivalent Representations Not surprisingly, these are an issue. But f -connectivity is very strong and we can use another nice theorem of GGHZ. Theorem if M is f -connected with f (0) = 0 and f (1) = 1, then M has a bounded number of inequivalent F- representations.

44 Inequivalent Representations Not surprisingly, these are an issue. But f -connectivity is very strong and we can use another nice theorem of GGHZ. Theorem if M is f -connected with f (0) = 0 and f (1) = 1, then M has a bounded number of inequivalent F- representations. This extends to strongly f -connected.

45 Corollary of Hypothesis 3 If k is very large, M 1 and M 2 are F-representable on the same ground set, M 1 is (f, k)-cpnnected, and X is the set of elements such that M 1 \x = M 2 \x for all x X, then X has bounded size.