The thesis consists of multiple problems from the field of extremal combinatorics. Chapter 1&2: Introduction and notations Chapter 3:We determine the minimal length of the longest trail in a fixed edge-density graph. This result is similar to the Erdos-Gallai theorem describing the maximal size of a graph not containing a path of length l. Chapter 4: We determine the extremal behaviour of 3-uniform k-fan-free hypergraphs. Chapter 5: The concept of almost intersecting set systems is introduced and multiple possible definitions (=l, <=l,uniform, non-uniform) are examined. Chapter 6: the concept of cross-Sperner pairs of families is introduced and analysed. Chapter 7: Further almost-properties are introduced and analysed ('almost Sperner', 'almost intersecting Sperner' and 'almost unrelated' families)