The assumption made in Euler's column theory is that all of the above options are correct, i.e., the failure of a column occurs due to buckling alone, the length of the column is very large as compared to its lateral dimensions, and the column material obeys Hooke's law.

1. Buckling as the cause of failure:
Euler's column theory assumes that the failure of a column is primarily due to buckling. Buckling refers to the sudden sideways deflection or bending of a slender column when subjected to compressive forces. It occurs when the compressive load exceeds the critical buckling load of the column. Euler's theory focuses on the structural behavior of columns under compressive loads and how they tend to buckle.

2. Long and slender columns:
Another assumption of Euler's column theory is that the length of the column is very large compared to its lateral dimensions. This assumption is based on the observation that long and slender columns are more prone to buckling than short and stout columns. When the length of a column is significantly larger than its cross-sectional dimensions, the applied compressive load can cause it to buckle due to the inability to resist lateral deflection.

3. Material behavior:
Euler's column theory assumes that the column material obeys Hooke's law. Hooke's law states that the stress-strain relationship in a material is linear within the elastic limit. In other words, the assumption assumes that the material of the column behaves elastically and the stress is directly proportional to the strain. This assumption allows for simplified calculations of the critical buckling load and the deflection of the column.

By considering all these assumptions together, Euler's column theory provides a simplified and practical approach to analyzing the buckling behavior of slender columns. It allows engineers to determine the critical buckling load and predict the failure mode of a column based on its length, cross-sectional dimensions, and material properties. However, it is important to note that Euler's theory is applicable only to columns with idealized conditions and certain limitations, and it may not accurately represent the behavior of real-world columns with complex geometries or non-linear material properties.