# Conjecture about minimal number of edge crossings in complete bipartite graphs

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.

One paper you might be interested in is Zarankiewiczʼs Conjecture is finite for each fixed $$m$$ 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 $$n$$.
It is a fascinating conjecture. The following might be a good reference for you: In 1997, Richter & Thomassen showed that $$\lim_{n\to\infty}cr(K_{n,n})\left(\begin{array}{c} n \\ 2 \end{array}\right)^{-2}$$ exists and is at most $$1/4$$. If the conjecture is true, the value of this limit is exactly $$1/4$$. (R.B. Richter, C. Thomassen, "Relations between crossing numbers of complete and complete bipartite graphs" Amer. Math. Monthly , 104 (1997) pp. 131–137)