The Maximum Product of Sizes of Cross-\({t}\)-Intersecting Families

Two families of sets [Formula: see text] and [Formula: see text] are called cross-[Formula: see text] -intersecting if [Formula: see text] for all [Formula: see text] and [Formula: see text]. Determining the maximum possible product of the sizes for such cross-[Formula: see text]-intersecting families is an active problem in extremal set theory. In this paper, we verify the following cross-[Formula: see text]-intersecting version of the Erdős–Ko–Rado theorem: for [Formula: see text] and [Formula: see text], the maximum value of [Formula: see text] for two cross-[Formula: see text]-intersecting families [Formula: see text] and [Formula: see text] is [Formula: see text]. Moreover, we characterize the extremal families attaining this bound. This result confirms a conjecture of Tokushige for [Formula: see text] and improves a lower bound established by Borg (J. Lond. Math. Soc., 2016) in this setting.