Arise, originate, emerge, to be established, to be founded, to be set up - these are all ways of translating the Slovak "vznik" or "vzniknúť", but of course it is good to know some typical combinations of words. Establish, found and set up are also equivalents of "založiť", especially for a company, firm, foundation, pension fund (very topical), but to translate "vzniknúť" you need the passive form (to be founded), because firms don't set themselves up, it's always the work of people.

Problems or conflict can arise or emerge - arise means more suddenly, emerge more gradually. Originate leads to the idea of where from, the origin of the problem, the need, or the idea - the cost-saving approach originated from (or in) the demand for improved results without increasing prices. Careful with "arise", because it is a strong verb - say "problems can arise", "an issue arose at the meeting", "new conflict has arisen in the coalition" - but that's no surprise.