Non-empty pairwise cross-intersecting families

Abstract: Two families

m a t h c a l A

and

m a t h c a l B

are cross-intersecting if

A c a p B e e m p t y s e t

for any

A i n m a t h c a l A

and

B i n m a t h c a l B

. We call

t

families

m a t h c a l A_{1}, m a t h c a l A_{2}, d o t s, m a t h c a l A_{t}

pairwise cross-intersecting families if

m a t h c a l A_{i}

and

m a t h c a l A_{j}

are cross-intersecting when

1 l e i < j l e t

. Additionally, if

m a t h c a l A_{j} e e m p t y s e t

for each

j i n [t]

, then we say that

m a t h c a l A_{1}, m a t h c a l A_{2}, d o t s, m a t h c a l A_{t}

are non-empty pairwise cross-intersecting. Let

m a t h c a l A_{1} s u b s e t [n] c h o o s e k_{1}, m a t h c a l A_{2} s u b s e t [n] c h o o s e k_{2}, d o t s, m a t h c a l A_{t} s u b s e t [n] c h o o s e k_{t}

be non-empty pairwise cross-intersecting families with

t g e q 2

,

k_{1} g e q k_{2} g e q c d o t s g e q k_{t}

, and

n g e q k_{1} + k_{2}

, we determine the maximum value of

s u m_{i = 1}^{t} | m a t h c a l A_{i} |

and characterize all extremal families. This answers a question of Shi, Frankl and Qian [Combinatorica 42 (2022)] and unifies results of Frankl and Tokushige [J. Combin. Theory Ser. A 61 (1992)] and Shi, Frankl and Qian [Combinatorica 42 (2022)]. The key techniques in previous works cannot be extended to our situation. A result of Kruskal-Katona is applied to allow us to consider only families