Keilim, Chapter 18, Mishnah 4. This chapter is occupied with beds and with the various pieces that were used together with a bed, and the recurring question is which of those pieces count as part of the bed itself and can therefore receive tumah, and which remain separate items that stay tahor. Our mishnah takes up one more such piece.
In the times of the Mishnah, thin strips of wood were sometimes set underneath the wooden frame of the bed, the malben. The purpose of these strips is not entirely clear to us. It may well be that they served to level the bed and to keep it from wobbling on an uneven floor. The question before us is whether those strips share the status of the bed and become tamei along with it.
The mishnah opens with the case: "Malben shenetano al leshonot", a bed frame that has been set to rest on leshonot, those thin wooden tongues or slats. What is the halachic standing of the slats underneath?
The first view: Rebbi Meir and Rebbi Yehudah rule "metam'in", meaning the slats do take on tumah. As they see it, the slats belong together with the frame and normally stay attached to it even when the bed is carried from place to place. Because they accompany the bed and are not separated in ordinary use, halachah counts them as part of the bed, so whatever tumah reaches the bed reaches them as well.
Rebbi Yose and Rebbi Shimon take the opposite view and rule "metaharin", the slats stay tahor. Their argument runs in reverse: there were occasions when the slats were lifted off the frame in order to transport the bed. Anything that gets detached in the routine handling of the bed is not reckoned as belonging to it, and therefore it does not receive tumah.
Rebbi Yose then buttresses his ruling with a comparison. He asks "mah shna zeh", what sets this case apart from the frames used by the Leviim, and he answers, "malbenei benei Levi tehorin", those frames of the Leviim are tahor. Because they were dismantled and taken away whenever they needed to be carried, they never took on the standing of a permanent component of the object, and so tumah does not apply to them. The identical reasoning should decide our case: since these slats too are removed on occasion, they should likewise remain tahor.
So the mishnah leaves us with a clean dispute resting on a single criterion: does the piece stay in place when the bed is carried, or is it taken off? Rebbi Meir and Rebbi Yehuda judge the strips to be permanently attached and thus part of the bed for tumah; Rebbi Yose and Rebbi Shimon judge them to be occasionally detached, and thus a separate item that stays tahor.