3. Extremal graph theory

How badly can Sidorenko fail for hypergraphs
Conjecture 3.1.
[Jacob Fox] Let $H$ be a 3uniform, 3partite hypergraph. Then any $n$vertex 3uniform hypergraph with $pn^3$ edges contains at least $p^{Ce(H)}n^{V(H)}$ copies of $H$.
For graphs, the bound $p^{e(H)+v(H)/2}n^{v(H)}$ is known. This might extend directly to hypergraphs. 
Expansion of nonSidorenko graphs
Given a $k$graph $F$, define the $r$expansion of $F$ to be the $r$graph obtained by enlarging each $k$edge of $F$ with a set of $rk$ vertices of degree one.Problem 3.2.
[Sam Spiro] Does there exist a nonSidorenko (hyper)graph $H$ whose $r$expansion is Sidorenko?
It is known that $C_5$ and its 3expansion are not Sidorenko. More generally, if the girth (length of the shortest loose cycle) of a hypergraph $F$ is odd, then $F$ is not Sidorenko. 
Induced packing and covering
Let $H$ be a fixed graph, $G$ an $n$vertex graph. If the minimum number of edges needed to add or remove from $G$ in order to remove all induced copies of $H$ is $o(n^2)$, then the regularity lemma implies that the maximum number of induced copies of $H$, with each pair intersecting in at most one vertex, is $o(n^2)$.Problem 3.3.
[Jacob Fox] Can we get better bounds, even for the case $H=C_5$? 
Turan exponents of hypergraphs
Conjecture 3.4.
[Jonathan Tidor] There exists $c>0$ such that the following holds. If $H$ is an $n$vertex, 3uniform hypergraph with at least $n^{3c/d}$ edges, then $H$ contains some/every 3uniform, 3partite linear hypergraph on $d$ vertices with at least $d^2/100$ edges.
Interesting cases are when $H$ is the set of triangles in some graph $G$, or a $p$random subset of triangles in $G$.
Cite this as: AimPL: Highdimensional phenomena in discrete analysis, available at http://aimpl.org/highdimdiscrete.