%I
%S 1,1,2,3,5,11,27,71,201,603,1905,6318,21887,78912,295260,1143536,
%T 4574144,18859678,80014835,348776577,1559776269,7147792819,
%U 33526120083,160785623546,787685471323,3938427356615,20082117944246,104349745809074,552166953567229
%N a(n) = ceiling(sqrt(n!)).
%C Axenovich's improvement to the Erdős strong Deltasystem conjecture. Erdős and Rado called a family of sets {A1, A2, .., Ak} a strong Deltasystem if all the intersections Ai INTERSECT Aj are identical, 1 <= i < j <= k. Denoting by f(n,k) the smallest integer m for which every family of nsets {A1, A2, .., Am} contains k sets forming a strong Deltasystem. Then Axenovich et al. proved f(n,3) < (n!)^((1/2) + epsilon)) < a(n) holds for every epsilon > 0, provided n is sufficiently large.  _Jonathan Vos Post_, Apr 29 2007; typos fixed by _Liyao Xia_, May 06 2014
%C For n>0, a(n) is the least m>0 such that n! <= m^2.  _Clark Kimberling_, Jul 18 2012
%H Chai Wah Wu, <a href="/A055228/b055228.txt">Table of n, a(n) for n = 0..807</a> (n = 0..300 from T. D. Noe)
%H M. Axenovich, D. FonDerFlaass and A. Kostochka, <a href="http://dx.doi.org/10.1016/0012365X(94)00185L">On set systems without weak 3Deltasubsystems</a>, Discrete Math. 138(1995), 5762.
%H Bela Bollobas, <a href="http://www.jstor.org/stable/2589077">To Prove and Conjecture: Paul Erdős and His Mathematics</a>, Am. Math. Monthly, 105 (March 1998)3, p. 232.
%H P. Erdős and R. Rado, <a href="http://www.renyi.hu/~p_erdos/196107.pdf">Intersection theorems for systems of finite sets I</a>, J. London Math. Soc. (2) 35(1960), 8590.
%H P. Erdős and R. Rado, <a href="http://www.renyi.hu/~p_erdos/196902.pdf">Intersection theorems for systems of finite sets II</a>, J. London Math. Soc. (2) 44(1969), 467479.
%F a(n) = A003059(A000142(n)).  _Jonathan Vos Post_, Apr 29 2007
%p A055228:=n>ceil(sqrt(n!)); seq(A055228(n), n=0..30); # _Wesley Ivan Hurt_, May 06 2014
%t Table[Ceiling[Sqrt[n!]], {n, 0, 30}] (* _Wesley Ivan Hurt_, May 06 2014 *)
%o (PARI) a(n) = ceil(sqrt(n!)) \\ _Michel Marcus_, Jul 30 2013
%Y Cf. A000142, A003059.
%K easy,nonn
%O 0,3
%A _Henry Bottomley_, Jun 21 2000
%E A comment stating that one of the terms was wrong has been deleted  the terms are correct.  _T. D. Noe_, Apr 22 2009
%E More terms from _Wesley Ivan Hurt_, May 06 2014
