Talmud – Chullin
Chullin 104b
The Talmud discusses the rabbinic stringency of not placing meat and milk on the same table, and records the debate about two people eating together — one meat, one dairy — and the concern that they may share food, requiring a reminder or partition.
מַתְנִי׳ הָעוֹף עוֹלֶה עִם הַגְּבִינָה עַל הַשּׁוּלְחָן, וְאֵינוֹ נֶאֱכָל, דִּבְרֵי בֵּית שַׁמַּאי. וּבֵית הִלֵּל אוֹמְרִים: לֹא עוֹלֶה וְלֹא נֶאֱכָל. אָמַר רַבִּי יוֹסֵי: זוֹ מִקּוּלֵּי בֵּית שַׁמַּאי וּמֵחוּמְרֵי בֵּית הִלֵּל. בְּאֵיזֶה שׁוּלְחָן אָמְרוּ? בְּשׁוּלְחָן שֶׁאוֹכֵל עָלָיו, אֲבָל בְּשׁוּלְחָן שֶׁסּוֹדֵר עָלָיו אֶת הַתַּבְשִׁיל – נוֹתֵן זֶה בְּצַד זֶה, וְאֵינוֹ חוֹשֵׁשׁ.
MISHNA: The meat of birds may be placed with cheese on one table but may not be eaten together with it; this is the statement of Beit Shammai. And Beit Hillel say: It may neither be placed on one table nor be eaten with cheese. Rabbi Yosei said: This is one of the disputes involving leniencies of Beit Shammai and stringencies of Beit Hillel. The mishna elaborates: With regard to which table are these halakhot stated? It is with regard to a table upon which one eats. But on a table upon which one prepares the cooked food, one may place this meat alongside that cheese or vice versa, and need not be concerned that perhaps they will be mixed and one will come to eat them together.
Why it matters — The foundational Talmudic sugya governing table use for meat and dairy, directly relevant to the tablecloth question.