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Can we have a civilization living at close to the speed of light?

Imagine many ships traveling close to the speed of light. If they diverge paths to explore distant galaxies, they could calculate their travel so that the same amount of time passes for all ships. Since traveling close to the speed of light will substantially slow down the passage of time for those on board the ship, this whole ensemble would allow many observers to simultaneously visit arbitrarily distant points in the universe in reasonable amount of time and without worrying about diverging time scale for those on different ships. I don't see any immediate holes in this idea but I wanted to see if anyone can spot some.