I don't think that's "X history more than Y history" a correct way to argue about history :-)
It's definitely a fair to name that Ukrainian history from your perspective. It's a also fair to name that Russian history.
A lot of people could argue that is specifically Cossack history, because Cossacks could be viewed as separate ethnicity with their own history.
Somebody will say that's Russian word is correct, because "Russian" is umbrella term for both velikorossy (Great Russians) and malorossy (Ukrainians) :-)
But it's definitely fair from modern Russian perspective and from modern Ukrainian perspective to claim this history as their own.
It's definitely a fair to name that Ukrainian history from your perspective. It's a also fair to name that Russian history.
A lot of people could argue that is specifically Cossack history, because Cossacks could be viewed as separate ethnicity with their own history.
Somebody will say that's Russian word is correct, because "Russian" is umbrella term for both velikorossy (Great Russians) and malorossy (Ukrainians) :-)
But it's definitely fair from modern Russian perspective and from modern Ukrainian perspective to claim this history as their own.