  Half graphs and their variants, such as ladders, semi-ladders and
co-matchings, are combinatorial objects that encode total orders in graphs.
Works by Adler and Adler (Eur. J. Comb.; 2014) and Fabia\'nski et al. (STACS;
2019) prove that in the powers of sparse graphs, one cannot find arbitrarily
large objects of this kind. However, these proofs either are non-constructive,
or provide only loose upper bounds on the orders of half graphs and
semi-ladders. In this work we provide nearly tight asymptotic lower and upper
bounds on the maximum order of half graphs, parameterized on the distance, in
the following classes of sparse graphs: planar graphs, graphs with bounded
maximum degree, graphs with bounded pathwidth or treewidth, and graphs
excluding a fixed clique as a minor.
  The most significant part of our work is the upper bound for planar graphs.
Here, we employ techniques of structural graph theory to analyze semi-ladders
in planar graphs through the notion of cages, which expose a topological
structure in semi-ladders. As an essential building block of this proof, we
also state and prove a new structural result, yielding a fully polynomial bound
on the neighborhood complexity in the class of planar graphs.
