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Computing the complexity of Markov bases is an extremely challenging problem;no formula is known in general and there are very few classes of toric idealsfor which the Markov complexity has been computed. A monomial curve $C$ in$\mathbb{A}^3$ has Markov complexity $m(C)$ two or three. Two if the monomialcurve is complete intersection and three otherwise. Our main result shows thatthere is no $d\in \mathbb{N}$ such that $m(C)\leq d$ for all monomial curves$C$ in $\mathbb{A}^4$. The same result is true even if we restrict to completeintersections. We extend this result to all monomial curves in $\mathbb{A}^n,n\geq 4$.

Unboundedness of Markov complexity of monomial curves in ${\mathbb A}^n$ for $n\geq 4$