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

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 ideals for 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 monomial curve is complete intersection and three otherwise. Our main result shows that there is no $din mathbb{N}$ 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 complete intersections. We extend this result to all monomial curves in $mathbb{A}^n, ngeq 4$.