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In \(\mathrm {PG}(2,q)\), the projective plane over the field \(\mathbf{F}_{q}\) of q elements, a (k, n)-arc is a set \(\mathcal {K}\) of k points with at most n points on any line of the plane. A fundamental question is to determine the values of k for which \(\mathcal {K}\) is complete, that is, not contained in a \((k+1,n)\)-arc. In particular, what are the smallest and largest values of k for a complete \(\mathcal {K}\), denoted by \(t_n(2,q)\) and \(m_n(2,q)\)? Here, a new lower bound for \(t_n(2,q)\) is established and compared to known values for small q.

A new lower bound for the smallest complete (k, n)-arc in $$\mathrm {PG}(2,q)$$ PG ( 2 , q )