Step 1: Analyze statement I.
Boron trihalides such as \(BF_3\), \(BCl_3\), \(BBr_3\), and \(BI_3\) are electron-deficient compounds.
However, they generally exist as monomeric molecules because back bonding from halogen to boron helps in reducing electron deficiency.
Therefore, the statement that boron trihalides can form dimeric structures is not correct in general.
So, statement I is incorrect.
Step 2: Analyze statement II.
Boron belongs to group 13 and its common stable oxidation state is \(+3\).
The \(+1\) oxidation state becomes more stable for heavier group 13 elements due to inert pair effect, especially in thallium.
Boron does not show \(+1\) as a stable oxidation state.
So, statement II is incorrect.
Step 3: Analyze statement III.
Boron is a second-period element and does not have vacant \(d\)-orbitals.
Therefore, it cannot expand its octet.
Its maximum covalency is four, as seen in species like:
\[
BF_4^-
\]
Thus, statement III is correct.
Step 4: Analyze statement IV.
Boron cannot form \(BF_6^{6-}\) because it cannot show coordination number \(6\).
Being a second-period element, boron can accommodate a maximum of only \(8\) electrons in its valence shell.
Hence, it can form \(BF_4^-\), but not \(BF_6^{6-}\).
So, statement IV is correct.
Step 5: Select the correct statements.
The correct statements are:
\[
III \text{ and } IV
\]
Therefore, the correct option is:
\[
(3)\; III,\;IV\;only
\]
Step 6: Final conclusion.
Hence, the correct answer is:
\[
\boxed{(3)\; III,\;IV\;only}
\]