the constraint becomes true. In this case we can restrict attention to the $2-$core. It is obtained by iteratively deleting all variables which occur at most once. Thus it is the maximal subformula in which each variable occurs at least twice.

We consider the random instance $F(n, p):$ Each equation over $n$ variables is picked independently with probability $p;$ the domain size $d$ and the number of slots per equation $k$ is fixed. We consider the case $p=c/n^{k-1}$ and the number of constraints is linear in $n$ whp. (with high probability, that is probability $1\,-\,o(1),  n$ large. ) The density of a formula is equal to the number of equations divided by the number of variables. The following is well known:

\[[@MOLLOY]\] \[Core\] 1. Conditional on the number of variables $n'$ and equations $m'$ of the $2-$core the $2-$core is a uniform random member of all formulas where each variable occurs at least twice.\
2. There exist $n'=n'(c)$ and $m'=m'(c)$ such that the number of variables of the $2-$core is $n'(1+o(1))$ and the number of equations $m'(1+o(1))$ whp.\
3. There exists a $T$ such that whp. for $c \le T-\varepsilon$ the $2-$core has density $\le 1-\varepsilon$ and for $c \ge  