  In the projective planes $\mathrm{PG}(2,q)$, we collect the smallest known
sizes of complete arcs for the regions \begin{align*} &\mbox{all }
q\le160001,~~ q \mbox{ prime power};\\ &Q_{4}=\{34 \mbox{ sporadic }q'\mbox{s
in the interval }[160801\ldots430007], \mbox{ see Table 3}\}. \end{align*}
  For $q\le160001$, the collection of arc sizes is complete in the sense that
arcs for all prime powers are considered. This proves new upper bounds on the
smallest size $t_{2}(2,q)$ of a complete arc in $\mathrm{PG}(2,q)$, in
particular \begin{align*} t_{2}(2,q)&<0.998\sqrt{3q\ln q}<1.729\sqrt{q\ln
q}&\mbox{ for }&&7&\le q\le160001;~~(1) \\
t_{2}(2,q)&<\sqrt{q}\ln^{0.7295}q&\mbox{ for }&&109&\le q\le160001;~~(2)\\
t_{2}(2,q)&<\sqrt{q}\ln^{c_{up}(q)}q,~~c_{up}(q)=\frac{0.27}{\ln q}+0.7,&\mbox{
for }&&19&\le q\le160001;~~(3)\\
t_{2}(2,q)&<0.6\sqrt{q}\ln^{\varphi_{up}(q;0.6)}
q,~~\varphi_{up}(q;0.6)=\frac{1.5}{\ln q}+0.802,&\mbox{ for }&&19&\le
q\le160001.~~(4) \end{align*} Moreover, the bounds (2) -- (4) hold also for
$q\in Q_{4}$. Also, \begin{align*} t_{2}(2,q)&<1.006\sqrt{3q\ln
q}<1.743\sqrt{q\ln q}&\mbox{ for }&&q\in Q_{4}.~~(5) \end{align*} Our
investigations and results allow to conjecture that the bounds (2) -- (5) hold
for all $q\geq109$.
