Introduction to Geometry. By H. S. M. Coxeter. Pp. xiv, 443. 1961. (John Wiley)

Counting copies of a fixed subgraph in F-free graphs

Some sharp results on the generalized Turán numbers

In a graph G , a geodesic between two vertices x and y is a shortest path connecting x to y . A subset S of the vertices of G is in general position if no vertex of S lies on any geodesic between two other vertices of S . The size of a largest set of vertices in general position is the general position number that we denote by g p ( G ) . Recently, Ghorbani et al. proved that for any k if n  ≥  k 3  −  k 2  + 2 k  − 2 , then g p ( K n n ,  k ) = ( n  − 1 choose k  − 1) , where K n n ,  k denotes the Kneser graph. We improve on their result and show that the same conclusion holds for n  ≥ 2.5 k  − 0.5 and this bound is best possible. Our main tools are a result on cross-intersecting families and a slight generalization of Bollobas’s inequality on intersecting set pair systems.

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On the general position problem on Kneser graphs

/pdf/generalized-turan-problems-for-even-cycles-34kkbbhue2.pdf

Generalized Turán problems for even cycles

Given a graph $H$ and a set of graphs $\mathcal F$, let $ex(n,H,\mathcal F)$ denote the maximum possible number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. We investigate $ex(n,H,\mathcal F)$ when $H$ and members of $\mathcal F$ are cycles. We determine the order of magnitude of $ex(n, C_{2l}, C_{2k})$ for any $l , k \ge 2$. Moreover, we determine $ex(n, C_{4}, C_{2k})$ asymptotically for all $k$. 
Solymosi and Wong proved that if Erd\H{o}s's Girth Conjecture holds, then for all $l \ge 3$, we have $ex(n, C_{2l}, \{ C_3, C_4, \ldots, C_{2l-2}\}) = \Theta(n^{2l/(l-1)})$. We prove that their result is sharp in the sense that if any other even cycle is also forbidden then the order of magnitude is smaller. More precisely, for any $k > l$, we have $$ex(n,C_{2l},\{C_3,C_4, \dots, C_{2l-2}\}\cup \{C_{2k}\})=\Theta(n^2).$$ 
We also determine the asymptotics of the number of $C_4$'s and the order of magnitude of the number of $C_6$'s for any given set of forbidden cycles.

Generalized Tur\'an problems for even cycles

An Erdős–Gallai type theorem for uniform hypergraphs

For a graph $F$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ (or a Berge-$F$ in short), if there is a bijection $f : E(F) \rightarrow E(\mathcal{H})$ such that for each $e \in E(F)$ we have $e \subset f(e)$ A hypergraph is Berge-$F$-free if it does not contain a Berge copy of $F$ We denote the maximum number of hyperedges in an $n$-vertex $r$-uniform Berge-$F$-free hypergraph by $\mathrm{ex}_r(n,\textrm{Berge-}F)$ 
In this paper we prove two general lemmas concerning the maximum size of a Berge-$F$-free hypergraph and use them to establish new results and improve several old results In particular, we give bounds on $\mathrm{ex}_r(n,\textrm{Berge-}F)$ when $F$ is a path (reproving a result of Győri, Katona and Lemons), a cycle (extending a result of Furedi and Ozkahya), a theta graph (improving a result of He and Tait), or a $K_{2,t}$ (extending a result of Gerbner, Methuku and Vizer) 
We also establish new bounds when $F$ is a clique (which implies extensions of results by Maherani and Shahsiah and by Gyarfas) and when $F$ is a general tree

Abhishek Methuku

Papers

Generalized Turán problems for even cycles

Generalized Tur\'an problems for even cycles

An Erdős–Gallai type theorem for uniform hypergraphs

General lemmas for Berge-Tur\'an hypergraph problems.

Asymptotics for the Turán number of Berge-K2,t