 Solution

For a fixed uniform hypergraph $H$ with $h$ vertices, uniformity $\ell$, and at least one edge, define its Turán density bywhere $\mathrm{ex}(n,H)$ is the maximum edge count of an $H$-free hypergraph. The limit exists: averaging the edge count over all $m$-vertex subsets of an extremal $n$-vertex hypergraph, with $n\ge m\ge \ell$, shows that the displayed normalized extremal numbers are nonincreasing. Thus their limit is their infimum.

For hypergraph supersaturation by sampling, choose $m\ge h$ so that $\mathrm{ex}(m,H)/\left(\genfrac{}{}{0ex}{}{m}{\ell }\right)\le \pi (H)+ϵ/2$. The expected edge count in a uniform $m$-vertex subset of $G$ is at least $(\pi (H)+ϵ)\left(\genfrac{}{}{0ex}{}{m}{\ell }\right)$. A subset containing no copy of $H$ has at most $(\pi (H)+ϵ/2)\left(\genfrac{}{}{0ex}{}{m}{\ell }\right)$ edges, while any subset has at most $\left(\genfrac{}{}{0ex}{}{m}{\ell }\right)$. Hence at least an $ϵ/2$ fraction of the subsets contain $H$. Every particular copy of $H$ lies in exactly $\left(\genfrac{}{}{0ex}{}{n-h}{m-h}\right)$ such subsets, so their number is at leastfor $n\ge N=\max (m,2h)$. TakeFor $n<N$, the requested integer lower bound is zero; for $n\ge N$, the calculation proves it. This positive $\delta$ depends only on $H$ and $ϵ$. If the density hypothesis is impossible there is nothing to prove. For an edgeless $H$, every $h$-set is already a copy and the conclusion follows directly.

For the edge-triangle objective, let $N(v)$ be the open neighbourhood of $v$ and define its local scoreIt counts the contribution to $k_2-c{k}_3$ of edges and triangles in a graph containing $v$. If $u,v$ are nonadjacent, making $u$ a false twin of $v$ changes the objective by $s(v)-s(u)$, since no edge or triangle in a graph uses both. More generally, partition vertices into false-twin classes, meaning equal open neighbourhoods. If distinct classes $U,V$ have no edges between them, making every vertex of $U$ a twin of a representative of $V$ changes the objective by $|U|(s(v)-s(u))$. Choose the direction with nonnegative change.

Among graphs maximizing the objective, choose one maximizing the sum of squared false-twin class sizes. The operation cannot split any old false-twin class and merges $U,V$, so it would strictly increase that secondary quantity. Therefore no two distinct classes can be nonadjacent. An objective maximizer is a complete multipartite graph. This edge-triangle symmetrization works for every real $c$, with no assumption on its sign.

For positive part sizes $a_1,\dots ,a_q$, the correct multipartite formula isThe first sum in the printed intermediate formula has inconsistent indices; the edge count is the pairwise-product sum above. Fix two part sizes with sum $A$ and let $B$ be the sum of all other sizes. All terms varying with the pair have the formChoose a maximizing partition with the fewest nonempty parts. If $1-cB\le 0$, merging those two parts cannot decrease $F$ and reduces the number of parts, a contradiction. Therefore this coefficient is positive for every pair. If two integer sizes differ by at least two, moving one vertex from the larger to the smaller increases their product and hence $F$, again a contradiction. All sizes differ by at most one, so a maximizer is a Turan graph. This is balancing a multipartite edge-triangle objective.

Next maximize the same polynomial on the compact simplex of nonnegative real sizes summing to $n$, starting with at least two possible parts. Choose a maximizer with the smallest positive support. The merging argument still applies, and now varying two positive unequal sizes towards equality improves the product. Thus every positive size is $n/q$ for some integer $q$. A two-part choice has value $n^2/4>0$ when $n>0$, so $q\ge 2$. The optimum is attained at rational sizes, and thereforeOne may take the initial simplex to have $\max (n,2)$ coordinates, so it includes every multipartite graph obtained above; zero coordinates are discarded.

Set $c=9/(4n)$. Dividing the right side by $n^2$ givesbecause $q$ is an integer at least two. Rearranging provesEquality in the continuous calculation occurs at two or three equal parts, explaining the triangle support line between bipartite and tripartite Turan graphs.