Paths of Length Three are $K_{r+1}$-Turán-Good

The generalized Turán problem ex$(n,T,F)$ is to determine the maximal number of copies of a graph $T$ that can exist in an $F$-free graph on $n$ vertices. Recently, Gerbner and Palmer noted that the solution to the generalized Turán problem is often the original Turán graph. They gave the name "$F$-Turán-good" to graphs $T$ for which, for large enough $n$, the solution to the generalized Turán problem is realized by a Turán graph. They prove that the path graph on two edges, $P_2$, is $K_{r+1}$-Turán-good for all $r \ge 3$, but they conjecture that the same result should hold for all $P_\ell$. In this paper, using arguments based in flag algebras, we prove that the path on three edges, $P_3$, is also $K_{r+1}$-Turán-good for all $r \ge 3$.

Vertex Turán problems for the oriented hypercube

In this short note we consider the oriented vertex Turán problem in the hypercube: for a fixed oriented graph F → \vec F , determine the maximum cardinality e x v ( F → , Q → n ) e{x_v}\left( {\vec F,{{\vec Q}_n}} \right) of a subset U of the vertices of the oriented hypercube Q → n {\vec Q_n} such that the induced subgraph Q → n [ U ] {\vec Q_n}\left[ U \right] does not contain any copy of F → \vec F . We obtain the exact value of e x v ( P k , → Q n → ) e{x_v}\left( {\overrightarrow {{P_k},} \,\overrightarrow {{Q_n}} } \right) for the directed path P k → \overrightarrow {{P_k}} , the exact value of e x v ( V 2 → , Q n → ) e{x_v}\left( {\overrightarrow {{V_2}} ,\,\overrightarrow {{Q_n}} } \right) for the directed cherry V 2 → \overrightarrow {{V_2}} and the asymptotic value of e x v ( T → , Q n → ) e{x_v}\left( {\overrightarrow T ,\overrightarrow {{Q_n}} } \right) for any directed tree T → \vec T .

Generalized Planar Turán Numbers

In a generalized Turán problem, we are given graphs $H$ and $F$ and seek to maximize the number of copies of $H$ in an $n$-vertex graph not containing $F$ as a subgraph. We consider generalized Turán problems where the host graph is planar. In particular, we obtain the order of magnitude of the maximum number of copies of a fixed tree in a planar graph containing no even cycle of length at most $2\ell$, for all $\ell$, $\ell \geqslant 1$. We also determine the order of magnitude of the maximum number of cycles of a given length in a planar $C_4$-free graph. An exact result is given for the maximum number of $5$-cycles in a $C_4$-free planar graph. Multiple conjectures are also introduced.

Generalizations of the Ruzsa–Szemerédi and rainbow Turán problems for cliques

Considering a natural generalization of the Ruzsa–Szemerédi problem, we prove that for any fixed positive integers r, s with r < s, there are graphs on n vertices containing $n^{r}e^{-O\left(\sqrt{\log{n}}\right)}=n^{r-o(1)}$ copies of K s such that any K r is contained in at most one K s . We also give bounds for the generalized rainbow Turán problem ex (n, H, rainbow - F) when F is complete. In particular, we answer a question of Gerbner, Mészáros, Methuku and Palmer, showing that there are properly edge-coloured graphs on n vertices with $n^{r-1-o(1)}$ copies of K r such that no K r is rainbow.