The question asks which statement is incorrect about the nature of carbon compounds. Let's analyze each option:

Option 1: Compounds of carbon, which are linked by only single bonds between the carbon atoms, are called saturated compounds. This statement is correct. Saturated carbon compounds have all single bonds between carbon atoms.

Option 2: Compounds of carbon are only formed with oxygen. This statement is incorrect. Carbon compounds can be formed with various elements such as hydrogen, nitrogen, sulfur, and halogens, not just oxygen.

Option 3: Compounds of carbon having double or triple bonds between their carbon atoms are called unsaturated compounds. This statement is correct. Unsaturated compounds contain at least one double or triple bond between carbon atoms.

Option 4: Carbon has the unique ability to form bonds with other atoms of carbon, giving rise to large molecules. This property is called catenation. This statement is correct. Catenation is the ability of carbon to form long chains and complex structures by bonding with other carbon atoms.

Thus, the incorrect statement is: Compounds of carbon are only formed with oxygen.