2024/08/20 by Carvalho, André, Machiavelo, António
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.2408.11213
We introduce some generalized topological concepts to deal with union-closed families, and show that one can reduce the proof of Frankl's conjecture to some families of so-called supratopological spaces. We prove some results on the structure of normalized families, presenting a new way of reducing such a family to a smaller one using dual families. Applying our reduction method, we prove a refinement of a conjecture originally proposed by Poonen. Finally, we show that Frankl's Conjecture holds for the class of families obtained from successively applying the reduction process to a power set.