Non-representable matroids and Ingleton's inequality

Question

Let $r$ be the rank function of a matroid. If the matroid is representable (over a field), then $r$ must satisfy Ingleton's inequalities. On the other hand, there are matroids that satisfy Ingleton's inequalities but are still not representable (already on $8$ elements such examples exist). However, not satisfying Ingleton's inequalities implies actually that for each $n\in\mathbb{N}$ the polymatroid $n\cdot r$ is not representable as well. This leads to the following questions:

-Are there matroids that satisfy Ingleton's inequalities such that $n\cdot r$ is not representable for all $n\in\mathbb{N}$?

-If yes, what are the smallest such examples? Are there such examples on $8$ elements?

Answer 1

I don't know the smallest such example, but here is an example with 14 elements.

• Let F be the Fano matroid: the 7-element matroid represented over any field of characteristic 2 by the set of non-zero vectors in $\{0,1\}^3$.
• Let N be the non-Fano matroid: the 7-element matroid represented over any field of characteristic not equal to 2 by the set of non-zero vectors in $\{0,1\}^3$.
• The direct sum of F and N has 14 elements, and its rank function satisfies Ingleton's inequality because it is a sum of two functions, each satisfying Ingleton's inequality.

In "Lexicographic Products and the Power of Non-Linear Network Coding", Anna Blasiak, Eyal Lubetzky, and I present an inequality that is satisfied by any positive scalar multiple of the rank function of a matroid representable in characteristic 2, but is violated by the rank function of N. This is inequality (6.8) in the paper. We also present an inequality that is satisfied by any positive scalar multiple of the rank function of a matroid representable in characteristic not equal to 2, but is violated by the rank function of F. This is inequality (6.17) in the paper. Since the direct sum of N and F violates both inequalities, the polymatroid defined by any positive scalar multiple of its rank function is not representable over a field of any characteristic.