What ukeire means
Ukeire (受け入れ), literally acceptance, is the set of tile types that improve a hand by lowering its shanten by one, that is, bringing it a step closer to tenpai. The term also refers to the raw count of still-unseen copies of those tiles. This count is an upper bound because an unseen copy may be in an opponent’s hand or the dead wall.
Comparing ukeire before a discard
You can measure ukeire at any shanten level, not only at tenpai. When choosing which tile to discard, efficiency-minded players compare the ukeire of each candidate and keep the shape with the widest acceptance. Once the hand reaches tenpai, its ukeire is simply the winning wait, and a solver reports both the accepted tile types and how many remain unseen.
Why the largest ukeire is not always best
With shanten, value and risk held comparable, wider ukeire improves expected speed: a ryanmen counts up to eight unseen copies while a kanchan or penchan counts up to four. Raw ukeire is not the whole story because it ignores value, yaku and safety. A completed hand without yaku cannot ron, and pushing a wide shape into danger can lead to a costly deal-in.