Forbidding \(K_{2,t}\) traces in triple systems

Abstract: Let

H

and

F

be hypergraphs. We say

H

contains

F

as a trace if there exists some set

S s u b s e t e q V (H)

such that

H |_{S} : = E c a p S : E i n E (H)

contains a subhypergraph isomorphic to

F

. In this paper we give an upper bound on the number of edges in a

3

-uniform hypergraph that does not contain

K_{2, t}

as a trace when

t

is large. In particular, we show that

l i m_{t o i n f t y} l i m_{n o i n f t y} f r a c m a t h r m e x (n, m a t h r m {T r}_{3} (K_{2, t})) t^{3 / 2} n^{3 / 2} = f r a c 16 .

Moreover, we show

f r a c 12 n^{3 / 2} + o (n^{3 / 2}) l e q m a t h r m e x (n, m a t h r m {T r}_{3} (C_{4})) l e q f r a c 56 n^{3 / 2} + o (n^{3 / 2})

.

Full work available at URL: https://arxiv.org/abs/2007.01827

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