  The Ramsey number $r(G,H)$ is the minimum $N$ such that every graph on $N$
vertices contains $G$ as a subgraph or its complement contains $H$ as a
subgraph. For integers $n \geq k \geq 1$, the $k$-book $B_{k,n}$ is the graph
on $n$ vertices consisting of a copy of $K_k$, called the spine, as well as
$n-k$ additional vertices each adjacent to every vertex of the spine and
non-adjacent to each other. A connected graph $H$ on $n$ vertices is called
$p$-good if $r(K_p,H)=(p-1)(n-1)+1$. Nikiforov and Rousseau proved that if $n$
is sufficiently large in terms of $p$ and $k$, then $B_{k,n}$ is $p$-good.
Their proof uses Szemer\'edi's regularity lemma and gives a tower-type bound on
$n$. We give a short new proof that avoids using the regularity method and
shows that every $B_{k,n}$ with $n \geq 2^{k^{10p}}$ is $p$-good.
  Using Szemer\'edi's regularity lemma, Nikiforov and Rousseau also proved much
more general goodness-type results, proving a tight bound on $r(G,H)$ for
several families of sparse graphs $G$ and $H$ as long as $|V(G)| < \delta
|V(H)|$ for a small constant $\delta > 0$. Using our techniques, we prove a new
result of this type, showing that $r(G,H) = (p-1)(n-1)+1$ when $H =B_{k,n}$ and
$G$ is a complete $p$-partite graph whose first $p-1$ parts have constant size
and whose last part has size $\delta n$, for some small constant $\delta>0$.
Again, our proof does not use the regularity method, and thus yields
double-exponential bounds on $\delta$.
