According to the post wwi territory map, which country was the "big brother" of syria?

France

The newly formed League of Nations, following World War I, set up "mandates" for governmental oversight of territories which were formerly under the control of the German Empire or the Ottoman Empire (the losing side in World War I). These mandates were in territories in Africa, the Middle East and Asia that were deemed not yet ready to govern themselves.  (There was some bias in that sort of judgment, of course.)  

In the Middle East, France was given the Mandate for Syria and Lebanon.  Britain was given the Mandate for Palestine as well as supervisory government over the Kingdom of Iraq.