The question of bounding the number of integer points on an elliptic curve $E/\mathbb{Q}$, shown to always be finite by Siegel, is an old question. There are various aspects to this problem. The aspect I am focusing on is the so-called repulsion principle of the integer points.
We first give some preliminaries. Suppose we have an elliptic curve $E/K$, $K$ a number field. Then the canonical height on $E$ admits a decomposition into local heights, given by
$$\displaystyle \hat{h}(P) = \frac{1}{[K : \mathbb{Q}]} \sum_{v \in M_K} d_v \lambda_v(P)$$
where $M_K$ is the set of places of $K$. A repulsion principle is then a statement of the form
$$\displaystyle \lambda_v(P_1 - P_2) \gg \min\{\lambda_v(P_1), \lambda_v(P_2)\},$$
where the exact statement depends on the place $v$, and of course depends on whether $v$ is an archimedean or non-archimedean place.
In this famous paper of Helfgott and Venkatesh, they proved that if $v$ is a place of potential good reduction for $E$, then we have
$$\displaystyle \lambda_v(P_1 - P_2) \geq \min\{\lambda_v(P_1), \lambda_v(P_2)\}.$$
If $v$ is a place of potential multiplicative (bad) reduction or if $v$ is archimedean, then one would need to partition the set of points into finitely many pieces, and some inequality of this sort holds on each piece; see Lemmas 3.1 - 3.3 in the aforementioned paper.
I am interested in a more basic version of this type of repulsion principle. Suppose we have an elliptic curve $E/\mathbb{Q}$, given by say a (minimal) short Weierstrass model:
$$\displaystyle E_{A,B} : y^2 = x^3 + Ax + B, \quad A,B \in \mathbb{Z},$$
and $p^4 | A \Rightarrow p^6 \nmid B$ for all primes $p$. Fixing this model, consider the set $E(\mathbb{Z})$ of integer points on this curve. More precisely, we have
$$E(\mathbb{Z}) = \{(x,y) \in \mathbb{Z}^2 : y^2 = x^3 + Ax + B\}.$$
Do we know of a statement of the form? That $E(\mathbb{Z})$ can be partitioned into $k = O(1)$ subsets $E_1 \cup \cdots \cup E_k$, such that whenever $(x_1, y_1), (x_2, y_2)$ are distinct elements of $E_j$, then
$$\displaystyle |x_1 - x_2| \geq f(A,B),$$
where $f(A,B)$ is some function that tends to infinity as $\max\{|A|,|B|\} \rightarrow \infty$?
