@article {IOPORT.06083217, author = {Balogh, J\'ozsef and Mubayi, Dhruv}, title = {Almost all triangle-free triple systems are tripartite.}, year = {2012}, journal = {Combinatorica}, volume = {32}, number = {2}, issn = {0209-9683}, pages = {143-169}, publisher = {J\'anos Bolyai Mathematical Society, Budapest; Springer-Verlag, Berlin}, doi = {10.1007/s00493-012-2657-4}, abstract = {For a fixed forbidden" $3$-graph $F$, $\text{Forb}(n,F)$ denotes the collection of 3-graphs with vertex-set $[n]=\{1,2,\dots,n\}$ which are $F$-free", i.e., which contain no copy of $F$. A 3-graph is {\it 3-partite\/} if its vertices can be partitioned into three disjoint parts so that all edges have exactly one vertex in each part; thus the maximum number of edges in a 3-partite 3-graph on $n$ vertices is $s(n):=\left\lfloor\frac{n}3\right\rfloor\left\lfloor\frac{n+1}3\right\rfloor\left\lfloor\frac{n+2}3\right\rfloor$; $T(n)$ denotes the number of 3-partite 3-graphs on $[n]$. $F_5$ denotes the 3-graph triangle", having vertices $\{1,2,3,4,5\}$ and edges $\{123,124,345\}$; evidently all 3-partite 3-graphs are $F_5$-free. Theorem 1: {\sl Almost all $F_5$-free 3-graphs on $[n]$ are 3-partite. More precisely, there is a constant $C>0$ such that $T(n)\le \left\vert \text{Forb}\left(n,F_5\right)\right\vert <\left(1+2^{Cn-\frac{2n^2}{45}}\right)T(n)$.\/} Corollary 1: {\sl As $n\to\infty$, $\log_2\left\vert \text{Forb}\left(n,F_5\right)\right\vert =s(n)+n\log_23+\Theta(\log n)$.\/} From the authors' abstract: Our proof uses the hypergraph regularity lemma of Frankl and R\"odl [{\it P.~Frankl and V.~R\"odl}, Random Struct. Algorithms 20, 131--164 (2002; Zbl 0995.05141)], and a stability theorem for triangle-free triple systems due to Keevash and the second author [{\it P.~Keevash and D.~Mubayi}, J. Comb. Theory, Ser. B 92, 163--175 (2004; Zbl 1052.05051)].}, reviewer = {William G. Brown (Montreal)}, identifier = {06083217}, }