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Problem 724

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Statement. Let f(n)f(n) be the maximum number of mutually orthogonal Latin squares of order nn. Is it true that

f(n)≫n1/2?f(n) \gg n^{1/2}?

Status. Open.

Source. erdosproblems.com/724, accessed 2026-09-04. Cite as: T. F. Bloom, Erdős Problem #724, https://www.erdosproblems.com/724.

References.

  • [BPS60] Bose, R. C. and Shrikhande, S. S. and Parker, E. T., Further results on the construction of mutually orthogonal Latin squares and the falsity of Euler's conjecture. Canadian J. Math. (1960), 189-203.
  • [Be83c] Beth, Thomas, Eine Bemerkung zur Abschätzung der Anzahl orthogonaler lateinischer Quadrate mittels Siebverfahren. Abh. Math. Sem. Univ. Hamburg (1983), 284-288.
  • [CES60] Chowla, S. and Erdős, P. and Straus, E. G., On the maximal number of pairwise orthogonal Latin squares of a given order. Canadian J. Math. (1960), 204-208.
  • [Wi74] Wilson, Richard M., Concerning the number of mutually orthogonal Latin squares. Discrete Math. (1974), 181-198.

Formalization. None recorded.

Progress

Not yet compiled.

Known Results

Not yet compiled.

Linked library material

These entries are derived from explicit links on library pages. They are navigation only and do not by themselves record mathematical progress.

Linked from (17)
Set Systems, Designs and Hypergraphsset_systems/abel_2024_improvements_lower_bounds_mutually_orthogonal_latinTheorem 1 (p. 3): there are 8 mutually orthogonal Latin squares of order 54Theorem 2 (p. 5): there are 10 mutually orthogonal Latin squares of order 96Theorem 3 (p. 6): there are 9 mutually orthogonal Latin squares of order 108set_systems/bose_1960_further_results_construction_mutually_orthogonal_latinTheorem 1 (p. 191): a pairwise balanced design of index unity with a clear set of equiblock components gives q* - 2 mutually orthogonal Latin squares of order vTheorem 10 (p. 202): two orthogonal Latin squares of every order v > 6Theorem 8 (p. 198): if k <= N(m) + 1, then N(km+1) and N(km+x), 1 < x < m, are bounded below by min(N(k), N(k+1), 1+N(m), ...) - 1Theorem 9 (p. 199): two orthogonal Latin squares of order 3m + 1 for every odd m, hence of every order 12t + 10set_systems/chowla_1960_maximum_number_pairwise_orthogonal_latin_squaresSection 2 (pp. 204--205): N(n) tends to infinityTheorem (p. 208): N(n) > (1/3) n^{1/91} for all n > n_0set_systems/wilson_1974_number_mutually_orthogonal_latin_squaresTheorem 2.3 (p. 186): if 0 <= u <= t then N(mt+u) >= min{N(m), N(m+1), N(t)-1, N(u)}Theorem 4.2 (p. 194): N(n) >= n^{1/17} - 2 for all n > n_0Theorem 5.1 (p. 196): N(n) >= 6 whenever n > 90
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