Upper Bounds on the Smallest Size of a Complete Arc in PG(2, q) under a Certain Probabilistic Conjecture

In the projective plane PG(2, q), we consider an iterative construction of complete arcs which adds a new point in each step. It is proved that uncovered points are uniformly distributed over the plane. For more than half of steps of the iterative process, we prove an estimate for the number of newly covered points in every step. A natural (and well-founded) conjecture is made that the estimate holds for the other steps too. As a result, we obtain upper bounds on the smallest size t_2(2, q) of a complete arc in PG(2, q), in particular, t2(2, q) <sqrt(q) sqrt(3 ln(q) + ln(ln(q)) + ln(3)) + sqrt(q/ (3 ln(q)))+ 3, t_2(2, q) < 1.87 sqrt(q ln q). Nonstandard types of upper bounds on t_2(2, q) are considered, one of them being new. Effectiveness of the new bounds is illustrated by comparing them with the smallest known sizes of complete arcs obtained in recent works of the authors and in the present paper via computer search in a wide region of q. We note a connection of the considered problems with the so-called birthday problem (or birthday paradox).