Eiruvin, Chapter 1, Mishnah 3. Our Mishnah discusses the required dimensions of the beam - the beam that serves as a marker and sign that the alleyway is not a public domain.
The wording of the Mishnah:
The beam they spoke about, meaning the one mentioned in the two previous Mishnayos, must meet two conditions:
"rechavah kedei lekabel ariach" - it must be wide enough to hold an ariach on top of it, something like a piece of stone or a brick.
"beri'ah kedei lekabel ariach" - it must be strong enough to actually support that ariach.
What is an ariach?
An ariach is half a brick, and a brick is generally three tefachim - so the width of an ariach is a tefach and a half. From here we learn that a beam one tefach wide is sufficient, since that is enough to hold the ariach and support it along its length.
In other words: an ariach that is a tefach and a half wide can be placed on a beam that is only one tefach wide, protruding a quarter of a tefach on each side, and this width is enough to hold it.
The Gemara's explanation:
The Gemara adds and explains how this was done in practice: the ariach was fixed in place with a bit of mortar placed underneath it to attach it, and for this purpose an overhang of a quarter of a tefach on each side was enough to secure the ariach on top of the beam. For this reason, a beam one tefach wide is sufficient.
In summary: In this Mishnah we learned the measurements of the beam that serves as a marker for an alleyway: it must be wide enough and strong enough to hold an ariach, which is half of a three-tefach brick, that is, a tefach and a half. The measurement of the beam was set at one tefach, since the ariach rests along its length and protrudes a quarter of a tefach on each side, and with the help of the mortar underneath it, it is secured and fixed in place on top of it.