The term 'building run-down' can be defined as a building that is in a state of disrepair. The crossword solution for this term is 'TENEMENT', which is a type of housing that is usually low-cost and often in a state of disrepair. Tenements are typically multi-family dwellings, meaning that multiple families live in the same building. Tenements are often found in urban areas and are known for their poor living conditions.