Consider the situation: You know that every $x$ that has property $P$ must have property $Q$. $Q$ is a rather strong condition but not strong enough to fulfill $P$. What is missing?
Consider the formulas of a first-order language, a distinguished set of axioms $S$, and the formulas with exactly one free variable, factored out by provable equivalence $P(x) \sim_S Q(x)$ iff $S \vdash P(x) \leftrightarrow Q(x)$ – for short: properties.
There is a partial order on the set of properties by $[P(x)] \Rightarrow_S [Q(x)]$ iff $S\vdash P(x) \rightarrow Q(x)$ – for short: $P \Rightarrow Q$.
Let $P \Rightarrow Q$, but $Q \not\Rightarrow P$, read: $Q$ is a proper necessary condition of $P$.
A property $D(x)$ may be called a defect of $Q(x)$ with respect to $P(x)$ if
- $D \not\Rightarrow P$
- $Q \wedge D \Rightarrow P$
A defect $D(x)$ may be called minimal if there is no other defect $D'(x)$ with $D(x) \Rightarrow D'(x)$.
Question #1: Is the search for (minimal) defects so manifest – in the working mathematician's life – that it is performed every day without needing a proper name? And is the notion ´– thus – of no (meta-)mathematical importance?
Question #2: ... or is there already a proper – and more common – name?
Question #3: ... or is the definition above and its presuppositions flawed?
The common divisor graph on the natural numbers shares one strong property $Q$ with the Rado (= random) graph: it contains any finite graph as an induced subgraph (which I guess is a quite general necessary condition for randomness). But it's not isomorphic to the Rado graph, i.e. does not fulfill its defining property $P$. I am now interested in the "defect" of the common divisor graph: which property $D$ - as minimal as possible - prevents the common divisor graph from being truly random?