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Let $P=\{p_{1},\ld,p_{r}\}\subset\Q[n_{1},\ld,n_{m}]$ be a family ofpolynomials such that $p_{i}(\Z^{m})\sle\Z$, $i=1,\ld,r$. We say that thefamily $P$ has {\it PSZ property} if for any set $E\sle\Z$ with$d^{*}(E)=\limsup_{N-M\ras\infty}\frac{|E\cap[M,N-1]|}{N-M}>0$ there existinfinitely many $n\in\Z^{m}$ such that $E$ contains a polynomial progression ofthe form \hbox{$\{a,a+p_{1}(n),\ld,a+p_{r}(n)\}$}. We prove that a polynomialfamily $P=\{p_{1},\ld,p_{r}\}$ has PSZ property if and only if the polynomials$p_{1},\ld,p_{r}$ are {\it jointly intersective}, meaning that for any $k\in\N$there exists $n\in\Z^{m}$ such that the integers $p_{1}(n),\ld,p_{r}(n)$ areall divisible by $k$. To obtain this result we give a new ergodic proof of thepolynomial Szemer\'{e}di theorem, based on the fact that the key to thephenomenon of polynomial multiple recurrence lies with the dynamical systemsdefined by translations on nilmanifolds. We also obtain, as a corollary, thefollowing generalization of the polynomial van der Waerden theorem: If$p_{1},\ld,p_{r}\in\Q[n]$ are jointly intersective integral polynomials, thenfor any finite partition of $\Z$, $\Z=\bigcup_{i=1}^{k}E_{i}$, there exist$i\in\{1,\ld,k\}$ and $a,n\in E_{i}$ such that$\{a,a+p_{1}(n),\ld,a+p_{r}(n)\}\sln E_{i}$.

Intersective polynomials and polynomial Szemeredi theorem