I am interested in the status of the conjecture about the minimum number of edge crossings $cr(K\_{m,n})$ in a drawing of the complete bipartite graph $K\_{m,n}$.


The Wikipedia article <https://en.wikipedia.org/wiki/Tur%C3%A1n%27s_brick_factory_problem> led me to study the original papers of Zarankiewicz (*On a problem of P. Turan concerning graphs*) from 1954 and of Urbanik (*Solution du problème posé par P. Turán*) from 1955.


I wondered whether someone could tell me whether an asymptotic approach has been successfully attempted (letting $n\to\infty$). If so, I would be very interested in any references for that.



Conjecture about minimal number of edge crossings in complete bipartite graphs
The Electronic Journal of Combinatorics has many [Dynamic Surveys](https://www.combinatorics.org/ojs/index.php/eljc/issue/view/Surveys) one of which is [The Graph Crossing Number and its Variants: A Survey](https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS21/pdf) by Schaefer which first appeared in 2013 and has been updated as recently as Feb 14, 2020. From the bottom of page 40 onto page 41 you will find this conjecture for complete bipartite graphs discussed (with many references). As far as I can tell from the survey the conjecture is open (both for exact values and asymptotically).


One paper you might be interested in is [Zarankiewiczʼs Conjecture is finite for each fixed $m$](https://doi.org/10.1016/j.jctb.2012.11.001) by Christiana, Richter, and Salazar. This paper shows that for each $m$ if the conjecture holds up for all $n$ up to some very large $N(m)$ (which is an explicit value), then the conjecture is true for $K\_{n,m}$ with any $n$.


This survey also references another survey [Turán’s Brick Factory Problem: The Status of the Conjectures of Zarankiewicz and Hill](https://link.springer.com/chapter/10.1007/978-3-319-31940-7_13) by Székely. (I haven't be able to access this survey so I don't exactly what's inside.)


