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For two graphs, G and F, and an integer r ≥ 2 we write G → (F)r if every r-coloring of the edges of G results in a monochromatic copy of F. In 1995, the first two authors established a threshold edge probability for the Ramsey property G(n, p) → (F)r, where G(n, p) is a random graph obtained by including each edge of the complete graph on n vertices, independently, with probability p. The original proof was based on the regularity lemma of Szemerédi and this led to tower-type dependencies between...

We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This affirmatively answers a question of Chung and Graham (2002) for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.

For an integer $k\ge 2$ and a $k$-uniform hypergraph $H$, let ${\delta }_{k-1}\left(H\right)$ be the largest integer $d$ such that every $(k-1)$-element set of vertices of $H$ belongs to at least $d$ edges of $H$. Further, let $t(k,n)$ be the smallest integer $t$ such that every $k$-uniform hypergraph on $n$ vertices and with ${\delta }_{k-1}\left(H\right)\ge t$ contains a perfect matching. The parameter $t(k,n)$ has been completely determined for all $k$ and large $n$ divisible by $k$ by Rödl, Ruci’nski, and Szemerédi in [, submitted]. The values of $t(k,n)$ are very close to $n/2-k$. In fact, the function $t(k,n)=n/2-k+c_{n,k}$, where $c_{n,k}\in \{3/2,2,5/2,3\}$ depends...