1. Complete hypergraphs
Let $K_m^k={[m]\choose k}$ be the complete $k$graph on $m$ vertices. Erd\H os offered a money prize for determining $\pi(K_m^k)$ for at least one pair $k,m$ with $m>k\ge 3$; the highest money value of the prize we found in the literature is \$3000 (Frankl and Füredi \cite[Page 323]{frankl+furedi:84}). It is still unclaimed. We also refer the reader to the survey by Sidorenko~\cite{sidorenko:95} that discusses $\pi(K_m^k)$ for some other $k,m$ in addition to those discussed below.
$K_4^3$ minus an edge
Let $K_4^$ be obtained from $K_4^3$ by removing one edge.Conjecture 1.4.
$\pi(K_4^)=\frac27$
The best known upper bounds come from flag algebra computations: Baber and Talbot [baber+talbot] (by using the method of Razborov [razborov:10] and generating a larger SDP program than that in [razborov:10]) showed that $\pi(K_4^)\le 0.2871$. 
There are many different constructions that achieve the lower bound (see Brown [brown:83], Kostochka [kostochka:82], and FonderFlaass [fonderflaass:88]), which is one of the reasons why this problem is so difficult. Successively better upper bounds were proved by de Caen [decaen:88], Giraud (see [chung+lu:01]), and Chung and Lu [chung+lu:01]. Razborov’s [razborov:10] flag algebra approach suggests that $\pi(K_4^3)\le 0.561...$ (and, if needed, this can be converted into a rigorous proof).

Remark. FonderFlaass [fonderflaass:88] presented a construction of $K_4^3$free graphs from digraphs. A weakening of Conjecture cj:K34 is that FonderFlaass’ construction cannot beat $\frac59$; some progress in this direction was made by Razborov [razborov:flaass].

Remark. Kalai [kalai:85:gc] (see also \cite[Section 11]{keevash:survey}) presented an interesting approach to $\pi(K_4^3)$.


Intro to this problem ...
Problem 1.3.
This problem is just a placeholder. Replace it bu a real problem, or we can jsut delete it later. 

Remark. A construction that achieves the lower bound can be found in \cite[Section 7]{sidorenko:95}. Mubayi and Keevash (see \cite[Section 9]{keevash:survey}) found a different construction (via digraphs).


Let us mention here another very interesting question for whose solution de Caen \cite[Page 190]{decaen:94} offered 500 Canadian dollars.
Problem 1.3.
Does $k(1\pi(K_{k+1}^k))$ tend to $\infty$ as $k\to\infty$?
Cite this as: AimPL: Hypergraph Turán problem, available at http://aimpl.org/hypergraphturan.