That's the thing about trademarks, they only cover the exact thing where it would be confusing, they don't cover all words that have any overlap.
You have been severely misinformed. A subset, something partial, something similar ... all would be subject to trademark violations. **If** the new company is conducting any activity in the same categories that the old trademark was filed under.
McAfee Associates doesn't exist!
The McAfee trademark is **currently** being used by Intel for various Intel Security products.
And the bare word "McAfee" does not prevent people named McAfee from using their name.
One a trademark has been issued there absolutely is. The only thing special about a name is that it is hard to trademark, **until** the name becomes closely associated with a product, service, company, etc. People named McDonald can't use their name for a restaurant, people named Disney can't use their name for an entertainment company, etc.
When the owners of that trademark bought the mark, they surely knew that there still existed a guy a named John McAfee, who sometimes does things to make money.
The buyers knew what I explained above. McAfee is not restrained from creating a new business, he just can't use his name on a new company, service, product etc in certain activities because **he** trademarked it and then **he** sold that trademark to someone else.