Forbidding $K_{2,t}$ Traces in Triple Systems

Abstract

Let $H$ and $F$ be hypergraphs. We say $H$ {\em contains $F$ as a trace} if there exists some set $S \subseteq V(H)$ such that $H|_S:=\{E\cap S: E \in E(H)\}$ 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
 $$\lim_{t\to \infty}\lim_{n\to \infty} \frac{\mathrm{ex}(n, \mathrm{Tr}_3(K_{2,t}))}{t^{3/2}n^{3/2}} = \frac{1}{6}.$$
 Moreover, we show $\frac{1}{2} n^{3/2} + o(n^{3/2}) \leqslant \mathrm{ex}(n, \mathrm{Tr}_3(C_4)) \leqslant \frac{5}{6} n^{3/2} + o(n^{3/2})$.

A note on the Turán number for the traces of hypergraphs

“…In this section, we generalise the result of Luo and Spiro for ex 3 (n, T r(C 4 )) in [4] by a similar analysis, and obtain an improved upper bound for ex 3 (n, T r(K 2,t )) when t is small.…”

“…In [3], Füredi and Luo also investigated the case when F is a star and gave both lower and upper bounds. Later, Luo and Spiro [4] gave an upper bound on the number of edges in a 3-uniform hypergraph which is T r(K 2,t )-free for t ≥ 14.…”

“…In this note, we first improve the lower bounds for ex r (n, T r(K 1,t )) and show that our lower bounds are optimal for infinitely many cases. Secondly, by adapting the method used in [4], we improve the upper bound for ex 3 (n, T r(K 2,t )) when t is small.…”

“…In this section, we focus on 3-uniform hypergraphs. In [4], Luo and Spiro gave the following theorem.…”

“…H\A to denote the hypergraph H after deleting some set of edges A from E(H). Following the notations in [4], for v ∈ V (H), we define the 1 and 2-neighborhood of v as…”