Hypergraphs

  1. Turán's conjecture: hypergraph Turán numbers: \(t_r(n, k)\) (Turán, not Erdős) ($1000)
  2. Exact values for \(t_3(n, 4) \) (Turán, not Erdős) ($500)
  3. Asymptotics of \(t_3(n, 5)\) (Turán, not Erdős)
  4. Adding an edge to extremal \(K^{(3)}_k\)-free graph gives two copies of \(K^{(3)}_k\)
  5. Adding an edge to extremal \(K^{(3)}_k\)-free graph gives a \(K^{(3)}_{k+1}\) missing an edge
  6. Avoiding triple systems (Brown, Sós)
  7. Lower bound for hypergraph Turán numbers
  8. Turán densities are rational
  9. Unavoidable stars (Rado)
  10. Special case: unavoidable \(3\)-stars
  11. Unavoidable stars of an n-set (Szemerédi)
  12. Weak \(\Delta\)-systems (Milner, Rado)
  13. Weak \(\Delta\)-systems of an n-set (Szemerédi)
  14. Erdős-Faber-Lovász conjecture: a simple hypergraph on \(n\) vertices has chromatic index at most \(n\) (Faber, Lovász)
  15. Minimum number of edges for a \(n\)-graph to not have Property B (that is: to not be \(2\)-colorable)
  16. Minimum number of edges for a \(3\)-chromatic \(4\)-graph
  17. Conjecture on \(3\)-chromatic hypergraphs (Lovász)
  18. Conjecture on minimum \(3\)-chromatic hypergraphs (Lovász)
  19. Maximum edges in a \(3\)-chromatic \(r\)-clique (Lovász)
  20. Maximum vertices in a \(3\)-chromatic \(r\)-clique (Lovász)
  21. \(3\)-chromatic cliques have edges with large intersection (Lovász) ($100)
  22. Number of sizes of edge intersections in a \(3\)-chromatic \(r\)-graph (Lovász)
  23. Jumping densities for \(3\)-hypergraphs ($500)
  24. Conjecture on covering a hypergraph (Lovász)
  25. Stronger conjecture on covering a hypergraph
  26. Maximum unavoidable hypergraphs (Chung)
  27. Unavoidable stars with fixed intersection size (Duke)
  28. Hypergraph decomposition (Chung, Graham)
  29. Characterize hypergraphs with maximum/minimum product of point and line covering numbers (Chung, Graham)
  30. Covering complete \(3\)-graphs (Tuza)
  31. \(r\)-sets with common union and intersection (Füredi)
  32. Finding small global vertex covers for r-graphs with small local vertex covers (Fon der Flaass, Kostochka, Tuza)
  33. A Ramsey-type conjecture for \(2\)-colorings of complete \(3\)-graphs
  34. A Ramsey-Turán upper bound for \(3\)-graphs
  35. A Ramsey-Turán lower bound for \(3\)-graphs (Sós)