Step 1: Set up the task.
We must choose the correct statements about the anomalous behaviour of boron from four given statements, numbered I to IV.
Step 2: Analyse statement I.
Statement I claims boron trihalides can form dimeric structures. Boron trihalides such as $BF_3$ and $BCl_3$ are generally monomeric because back bonding from the halogen lone pairs into the empty boron $p$ orbital relieves the electron deficiency. So statement I is incorrect.
Step 3: Analyse statement II.
Statement II claims boron shows $+1$ as a stable oxidation state. Boron is in group 13 and its characteristic stable oxidation state is $+3$, not $+1$. The $+1$ state is significant only for the heavier members through the inert pair effect. So statement II is incorrect.
Step 4: Analyse statement III.
Statement III claims the maximum covalency of boron is four. Boron has only four available orbitals ($2s$ and three $2p$), so it cannot exceed a covalency of four, as in $BF_4^-$. So statement III is correct.
Step 5: Analyse statement IV.
Statement IV claims boron does not form $BF_6^{3-}$ ion. Because boron lacks $d$ orbitals and is limited to a covalency of four, it cannot expand to six bonds, so it does not form such an ion. Statement IV is correct.
Step 6: State the answer.
The correct statements are III and IV only, matching the key.
\[ \boxed{III,\; IV \text{ only}} \]