Let $\Delta$ be a finite simplicial complex on $n$ vertices. Let $S = \mathbb{k}[\mathbf{x}]$ be a polynomial ring over a field $\mathbb{k}$ in $n$ variables and $I$ be the Stanley-Reisner ideal of $\Delta$ in $S$. It is well-known that the Betti numbers (graded or otherwise) appearing in the free resolution of $I$ may depend on the ground field $\mathbb{k}$. A classic example of this is the minimal triangulation of the real projective plane with Stanley-Reisner ideal $$I = \langle abc, abe, acf, ade, adf, bcd, bdf, bef, cde, cef \rangle.$$ When the characteristic of $\mathbb{k}$ is $2$ then the projective dimension of $I$ (as an $S$-module) is 3 while in all other cases it is 2. This example and others can be found in Dalili and Kummini's paper Dependence Of Betti Numbers On Characteristic and the references there.

In all of the examples I've seen of complexes whose Betti numbers are characteristic-dependent, the offending characteristic has always been 2, and only 2. That is, the Betti numbers are stable in all cases except when the characteristic of $\mathbb{k}$ is 2. My intuition tells me there must be ideals with other offending characteristics. As I have no idea how to construct one and my computer experiments (using RandomIdeals) have come up empty my questions are the following.

**Question** 1: Are there known examples of square-free monomial ideals with characteristic-dependent Betti numbers for characteristics other than two?

**Question** 2: Is there one whose only offending characteristic is 3?

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