Explanation of notation required? What is the meaning of the notation
$sign(k \mod 4)$
where $k$ is a positive integer?  The notation can be found, for example in the paper "Minimum triangle-free graphs".
 A: $k\bmod 4$ is the remainder obtained when $k$ is divided by $4$ (for example, $14\bmod 3$ is $2$).
The sign function  ($sign$, as you have it)  is $1$ for positive arguments, $-1$ for negative arguments, and $0$ if the argument is $0$. That is:
$$
{\rm sign}\,(x)=\cases{\hphantom{-}1,& x>0\cr \hphantom{-}0,&x=0\cr -1,& x<0 }\ \ .
$$
So, ${\rm sign}\,(k\bmod4)$ will be $0$ if $k$ is divisible by $4$ and $1$ otherwise. 
A: As is often the case, one can easily deduce the meaning from its use. On p. 88 we find 

Reading the proof we infer that  for $\rm\ x\ =\ k\ mod\ 4\ $  
$\rm\qquad\qquad\qquad x\ =\ 0\ \ \Rightarrow\ \ sign(x)\ =\ 0$  
$\rm\qquad\qquad\qquad x\ \ne\ 0\ \ \Rightarrow\ \ sign(x)\ =\ 1$
So the author requires that, $\rm\ mod\ 4,\:$ the $\rm\:sign\:$ map fixes $0$ and maps all  nonzero residues to $\rm\:1\:.\:$  
Note that if the $\rm\:mod\ 4\:$ map uses a balanced residue system $\rm\:\{-1, 0, 1, 2\}\ $ then the $\rm\:sign\:$ map is not that given in David's answer, since it has $\rm\:sign(-1) = -1\:,\:$ but the author requires $\rm\:sign(-1) = 1\:.$
