in Tic-tac-toe on an NxN board, what is the minimum goal (number in a row) that guarantees a tie?
This generalization of Tic-Tac-Toe is called m,n,k-game. (the goal is to get k in a row on a (m,n) board).

Some known bounds: (source wikipedia)


(5,5,4) is a draw. 
(6,6,5) is a draw. 
(7,7,5) and (8,8,5) are draws.   
(15,15,5) is a win. 
(9,6,6) and (7,7,6) are both draws via pairings.


When the goal is 9 or larger (k>=9) the second player can force a draw:  


  When k = 9 and the board is infinite, the second
  player can draw via a "pairing strategy". A draw on an infinite board
  means that the game will go on forever with perfect play. A pairing
  strategy involves dividing all the squares of the board into pairs in
  such a way that by always playing on the pair of the first player's
  square, the second player is ensured that the first player cannot get
  k in a line. A pairing strategy on an infinite board can be applied to
  any finite board as well - if the strategy calls for making a move
  outside the board, then the second player makes an arbitrary move
  inside the board.