A hypergraph Tur\'an theorem via lagrangians of intersecting families

Let \mc{K}_{3,3}^3 be the 3-graph with 15 vertices $\{x_i, y_i: 1 \le i \le 3\}$ and $\{z_{ij}: 1 \le i,j \le 3\}$, and 11 edges $\{x_1, x_2, x_3\}$, $\{y_1, y_2, y_3\}$ and $\{\{x_i, y_j, z_{ij}\}: 1 \le i,j \le 3\}$. We show that for large $n$, the unique largest \mc{K}_{3,3}^3-free 3-graph on $n$ vertices is a balanced blow-up of the complete 3-graph on 5 vertices. Our proof uses the stability method and a result on lagrangians of intersecting families that has independent interest

Hitting time results for Maker-Breaker games

We study Maker-Breaker games played on the edge set of a random graph. Specifically, we consider the random graph process and analyze the first time in a typical random graph process that Maker starts having a winning strategy for his final graph to admit some property \mP. We focus on three natural properties for Maker's graph, namely being $k$-vertex-connected, admitting a perfect matching, and being Hamiltonian. We prove the following optimal hitting time results: with high probability Maker wins the $k$-vertex connectivity game exactly at the time the random graph process first reaches minimum degree $2k$; with high probability Maker wins the perfect matching game exactly at the time the random graph process first reaches minimum degree $2$; with high probability Maker wins the Hamiltonicity game exactly at the time the random graph process first reaches minimum degree $4$. The latter two statements settle conjectures of Stojakovi\'{c} and Szab\'{o}.Comment: 24 page

Universality of graphs with few triangles and anti-triangles

We study 3-random-like graphs, that is, sequences of graphs in which the densities of triangles and anti-triangles converge to 1/8. Since the random graph ${\mathcal G}_{n,1/2}$ is, in particular, 3-random-like, this can be viewed as a weak version of quasirandomness. We first show that 3-random-like graphs are 4-universal, that is, they contain induced copies of all 4-vertex graphs. This settles a question of Linial and Morgenstern. We then show that for larger subgraphs, 3-random-like sequences demonstrate a completely different behaviour. We prove that for every graph $H$ on $n\geq R(10,10)$ vertices there exist 3-random-like graphs without an induced copy of $H$. Moreover, we prove that for every $\ell$ there are 3-random-like graphs which are $\ell$-universal but not $m$-universal when $m$ is sufficiently large compared to $\ell$.Comment: 12 page

On the inducibility of cycles

In 1975 Pippenger and Golumbic proved that any graph on $n$ vertices admits at most $2e(n/k)^k$ induced $k$-cycles. This bound is larger by a multiplicative factor of $2e$ than the simple lower bound obtained by a blow-up construction. Pippenger and Golumbic conjectured that the latter lower bound is essentially tight. In the present paper we establish a better upper bound of $(128e/81) \cdot (n/k)^k$. This constitutes the first progress towards proving the aforementioned conjecture since it was posed