![](/files/happy5.png)
Turan, Paul
Turán's problem for hypergraphs ★★
Author(s): Turan
Conjecture Every simple
-uniform hypergraph on
vertices which contains no complete
-uniform hypergraph on four vertices has at most
hyperedges.
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ 3n $](/files/tex/f36625fb2086623f510fdcab4f53cd27a419800a.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ \frac12 n^2(5n-3) $](/files/tex/77fae1e209cf2ef781abf9d88573e4981aa00b15.png)
Conjecture Every simple
-uniform hypergraph on
vertices which contains no complete
-uniform hypergraph on five vertices has at most
hyperedges.
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ 2n $](/files/tex/56259815f2fdf87e92dd22e0058206e8e20fb986.png)
![$ 3 $](/files/tex/4aaf85facb6534fd470edd32dbdb4e28f6218190.png)
![$ n^2(n-1) $](/files/tex/edaf55f28dcd60182c7c9d88572ea01709769bde.png)
Keywords:
The Erdos-Turan conjecture on additive bases ★★★★
Let . The representation function
for
is given by the rule
. We call
an additive basis if
is never
.
Conjecture If
is an additive basis, then
is unbounded.
![$ B $](/files/tex/4369e4eb2b0938fb27436a8c4f4a062f83d4d49e.png)
![$ r_B $](/files/tex/9503779c20e2cdbf39f574df5eb6379cb82922d7.png)
Keywords: additive basis; representation function
The Crossing Number of the Complete Bipartite Graph ★★★
Author(s): Turan
The crossing number of
is the minimum number of crossings in all drawings of
in the plane.
Conjecture
![$ \displaystyle cr(K_{m,n}) = \floor{\frac m2} \floor{\frac {m-1}2} \floor{\frac n2} \floor{\frac {n-1}2} $](/files/tex/aecee36502739c16e07eea1fc64a43a9e95e9374.png)
Keywords: complete bipartite graph; crossing number
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