I want to save a thread of messages from someone, but I don't want them on my iPhone, only on my Mac.
If I delete the thread on my iPhone, will that also delete it from Messages on my Mac?
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As of iOS 10, conversation threads in the Messages app are not synced across devices. So deleting a thread on one device will NOT affect other devices. You can safely delete this conversation from your iPhone, and it will stay on your Mac.
Do note that coming in iOS 11 is iCloud Messages, which will synchronize your text conversations across devices. If you enable this feature, previously deleted conversations from one device will return to that device if they still exist on another. After that, deleting a conversation anywhere will delete it everywhere.
You will, of course, have the option to leave that feature disabled. In that case, conversations will keep behaving as they did in iOS 10—independently on each device.