Fractal Classes of Matroids

FRACTAL CLASSES OF MATROIDS DILLON MAYHEW, MIKE NEWMAN, AND GEOFF WHITTLE Abstract. A minor-closed class of matroids is (strongly) fractal if the number of n-element matroids in the class is dominated by the number of n-element excluded minors. We conjecture that when K is an infinite field, the class of K-representable matroids is strongly fractal. We prove that the class of sparse paving matroids with at most k circuit- hyperplanes is a strongly fractal class when k is at least three. The minor-closure of the class of spikes with at most k circuit-hyperplanes (with k ≥ 5) satisfies a strictly weaker condition: the number of 2t-element matroids in the class is dominated by the number of 2t-element excluded minors. However, there are only finitely many excluded minors with ground sets of odd size. Dedicated to Joseph Kung, in gratitude for all his contributions to the matroid community. 1. Introduction In [5] we proved that every real-representable matroid is contained as a minor in an excluded minor for the class of real-representable matroids. (The same phenomenon holds for any infinite field.) Geelen and Camp- bell strengthened this by showing that every real-representable matroid is a minor of a complex-representable excluded minor for the class of real- representable matroids [1]. In contrast to these results, the resolution of Rota's conjecture [2] implies that there are only finitely many excluded minors for F-representability when F is a finite field. In this article we consider another possible dichotomy between finite fields and infinite fields in matroid representation theory. Let M be a minor-closed class of matroids, and let EX be the class of excluded minors for M. For any non-negative integer n, let mn be the number of non-isomorphic n-element matroids in M. Thus mn counts the n-element members of M modulo the equivalence relation of isomorphism. Similarly, let xn be the number of non-isomorphic n-element matroids in EX . (Henceforth, when we refer to a matroid, we usually mean an isomorphism class of matroids.) We consider the probability that a matroid chosen randomly from the n-element members of M[EX is an excluded minor. In other words, we consider the ratio x n : mn + xn Date: November 21, 2019. 1 2 MAYHEW, NEWMAN, AND WHITTLE We will denote this fraction by ΓM(n). We consider only the case that M contains infinitely many matroids, so that ΓM(n) is defined for all n. If M has only finitely many excluded minors, then ΓM(n) = 0 for all large enough values of n. It is impossible for ΓM(n) to be equal to one, but it could tend to one in the limit. In this case our random choice is asymptotically certain to be an excluded minor, so in some sense, the class M is eventually overwhelmed by its `boundary': the set of excluded minors. This leads us to the following terminology. Definition 1.1. Let M be a minor-closed class of matroids. If lim ΓM(n) = 1; n!1 then M is a strongly fractal class. If M is the class of F-representable matroids where F is a finite field, then ΓM(n) = 0 for all large enough values of n, since Rota's conjecture is true. We believe that this fails for infinite fields in the strongest possible way. Conjecture 1.2. Let K be an infinite field. The class of K-representable matroids is strongly fractal. The class of gammoids is like the class of real-representable matroids, in that the excluded minors form a maximal antichain [4]. We conjecture that the class of gammoids is strongly fractal. In the present article we are content merely to establish the non-obvious fact that strongly fractal classes exist. A matroid is sparse paving if every non-spanning circuit is a hyperplane. Theorem 1.3. Let k ≥ 3 be a positive integer. Let Pk be the class of sparse paving matroids with at most k circuit-hyperplanes. Then Pk is strongly fractal. A rank-r spike has a ground set of size 2r, say fa1; b1; : : : ; ar; brg. Assume that r > 3. Then the non-spanning circuits are exactly the sets of the form fai; bi; aj; bjg, along with (possibly) some sets that intersect each fai; big in either ai or bi. Any circuit of the latter type is also a hyperplane. Sometimes such a matroid is called a tipless spike. The class of spikes with a bounded number of circuit-hyperplanes is not minor-closed, but if we close it under minors, we obtain a class that has a weaker fractal property. Definition 1.4. Let M be a minor-closed class of matroids. If ΓM(1); ΓM(2); ΓM(3);::: contains an infinite subsequence that converges to one, then M is weakly fractal. Obviously a strongly fractal class is weakly fractal. Theorem 1.5. Let k ≥ 5 be an integer. Let Sk be the class produced by taking minors of the spikes with at most k circuit-hyperplanes. Then Sk is a weakly fractal class, but is not strongly fractal. FRACTAL CLASSES OF MATROIDS 3 The reason the class in Theorem 1.5 is not strongly fractal is that eventually there are no excluded-minors with odd-cardinality ground sets (Lemma 3.10). In other words, ΓSk (2t + 1) = 0 for all large-enough values of t. However ΓSk (2t) converges to one. Definition 1.4 does not require the subsequence that converges to one to have any particular structure. However, it is inconceivable that the following conjecture fails. Conjecture 1.6. Let M be a weakly fractal class of matroids. There exist integers a and b such that the sequence ΓM(a + b); ΓM(2a + b); ΓM(3a + b);::: converges to one. In the case of the class Sk, the integers a = 2 and b = 0 satisfy Conjec- ture 1.6. The only fractal classes we know of contain infinite antichains, and we think this exemplifies a general pattern. Conjecture 1.7. Any weakly fractal class of matroids contains an infinite antichain. We close this introduction with a consequence of the existence of strongly fractal classes. Proposition 1.8. Let γ be a real number in [0; 1]. Let M0 be a strongly fractal class of matroids. There exists a minor-closed class of matroids, M ⊇ M0, such that limn!1 ΓM(n) = γ. Proof. For any positive integer i, let N(i) be the smallest integer such that ΓM0 (n) ≥ (i − 1)=i for every n ≥ N(i). The existence of N(i) follows from limn!1 ΓM0 (n) = 1. Note that N(1);N(2);N(3);::: is a non-decreasing sequence. We construct a series of classes, M0 ⊆ M1 ⊆ M2 ⊆ · · · with the following properties. Let n be a positive integer, and choose i so that N(i) ≤ n ≤ N(i + 1). Then the only matroids in Mn − Mn−1 are n-element excluded minors for Mn−1, and γ − 1=i ≤ ΓMn (n) ≤ γ. Note that the first condition implies that Mn is a minor-closed class. Assume that we have successfully constructed Mn−1. Choose i so that N(i) ≤ n ≤ N(i + 1). Let mn and xn respectively denote the number of n-element matroids in M0, and the number of n-element excluded minors for M0. Because ΓM0 (n) ≥ (i − 1)=i, it follows that mn + xn ≥ i, for otherwise the only way to satisfy ΓM0 (n) ≥ (i − 1)=i is for mn to be zero. This is impossible because M0 must contain infinitely many matroids. From mn + xn ≥ i and ΓM0 (n) ≥ (i − 1)=i, we in turn derive xn ≥ i − 1. Note that an n-element excluded minor for M0 is not in Mn−1, but all of its proper minors are in M0. Therefore any such matroid is an excluded minor 0 for Mn−1. Let xn be the number of n-element excluded minors for Mn−1, 0 where we have just demonstrated that xn ≥ xn ≥ i − 1. 4 MAYHEW, NEWMAN, AND WHITTLE We observe that 0 xn xn i − 1 0 ≥ ≥ : mn + xn mn + xn i Let z be the smallest integer such that 0 xn − z 0 ≤ γ; mn + xn 0 and note that z is in f0; 1; : : : ; xng. Now we choose z matroids from the n-element excluded minors for Mn−1, and construct Mn by adding these z matroids to Mn−1. The n-element members of Mn are exactly the n-element members of M0 along with the z matroids we have added. The n-element excluded minors 0 for Mn are the xn − z excluded minors that we did not add to Mn−1. Therefore 0 xn − z ΓMn (n) = 0 ≤ γ: (mn + z) + (xn − z) 0 Furthermore, since mn + xn ≥ i, our choice of z means that ΓMn (n) is at least γ − 1=i. In this way we can construct the entire sequence M0 ⊆ M1 ⊆ M2 ⊆ · · · . Now we define M to be the union [n≥0Mn. The n-element members of M are the n-element members of Mn, and the n-element excluded minors for M are the n-element excluded minors for Mn. Therefore ΓM(n) = ΓMn (n). Since this value is between γ − 1=i and γ, where N(i) ≤ n ≤ N(i + 1), it follows that ΓM(n) converges to γ, as desired. We use P(X) to denote the power set of the set X. The symbol Z≥0 stands for the set of non-negative integers.

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