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An $n$-vertex graph $G$ is $(\rho, d)$-dense if for all $U \subseteq V(G)$ of size at least $\rho n$, the induced subgraph $G[U]$ contains at least $\frac{d}{2}|U|^2$ edges. In 2010, partly motivated by the theory of quasirandomness, Kohayakawa, Nagle, R\"{o}dl, and Schacht conjectured that, for any fixed graph $F$, th...
We denote by $f_k(\Gamma)$ the largest integer with the property that every subset of a finite abelian group $\Gamma$ of size at least $|\Gamma| - f_k(\Gamma)$ is a $k$-fold sumset. Extending a recent result of Alon and Pham, we prove that $$ f_k(\Gamma) \leq \widetilde{O} \left(n^{(2k-1)/(4k-3)}\right) $$ holds for al...
The Kahn--Lov\'{a}sz theorem gives a sharp upper bound on the number of perfect matchings in a graph in terms of its degree sequence, extending the classical Br\'{e}gman--Minc inequality for bipartite graphs. In this paper, we establish an asymptotically sharp extension of the Kahn--Lov\'{a}sz theorem to $F$-factors fo...
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