The main reference is:

Major open problems in combinatorics include:

  1. Number of Magic squares (sequence A006052 in OEIS)
  2. Finding a formula for the probability that two elements chosen at random generate the symmetric group
  3. Frankl’s union-closed sets conjecture: for any family of sets closed under sums there exists an element (of the underlying space) belonging to half or more of the sets See wikipedia entry
  4. The Lonely runner conjecture: if runners with pairwise distinct speeds run round a track of unit length, will every runner be “lonely” (that is, be more than a distance from each other runner) at some time?
  5. Singmaster’s conjecture: is there a finite upper bound on the multiplicities of the entries greater than 1 in Pascal’s triangle?
  6. The 1/3-2/3 conjecture: does every finite partially ordered set contain two elements x and y such that the probability that x appears before y in a random linear extension is between 1/3 and 2/3?

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