Worst-case ratios for connected bipartite Turán numbers

Determine exactly, or improve the lower bounds obtained for, the worst-case ratios Y_{b,c} and Y_{b+c} associated with connected bipartite Turán numbers of trees.

Background

The paper introduces asymptotic ratios comparing connected and unrestricted bipartite Turán numbers for trees. In particular, Y_{b,c} measures the asymptotic minimum normalized value of ex_{b,c}(n,T), while Y_{b+c} compares connected and unrestricted bipartite Turán numbers. The authors establish the exact value Y_b = 2/3 and provide lower bounds for Y_{b,c} and Y_{b+c}, but do not determine these latter quantities exactly.

Problem 13 explicitly asks for exact values or improved lower bounds, making this an unresolved asymptotic extremal problem arising from the paper’s general study of bipartite Turán numbers for trees.

References

Perhaps the most important open problems emerging from this paper, in this initial stage, are: Problem 13. Determine exactly or improve upon the lower bounds obtained for Yb,c and Yb+c.

— Bipartite Turán number of trees  (2502.09052 - Caro et al., 13 Feb 2025) in Problem 13, Section 4, “Concluding Remarks and Open Problems”