The Eventown theorem states that a family of subsets of an 
-element
set has at most 
members if each member and the intersection
of every two distinct members have even size. Here 
is the floor function.
Equality is attained by taking all unions of a fixed collection
of disjoint pairs (Banerjee and Dewan 2026).
Banerjee and Dewan (2026) extended the theorem to multisets on 
elements with each multiplicity at most 
. The multiplicity of an element in an intersection
is the minimum of its two multiplicities, and size counts all repetitions. If 
is the maximum size of a family
with even member and pairwise-intersection cardinalities, then
 |
and
 |
where 
,
in the even case, and 
in the odd case. The case 
recovers the set theorem. Banerjee and Dewan (2026) also
characterized all maximum families for 
.
