The independence number of non-uniform uncrowded hypergraphs and an anti-Ramsey type result

We prove the following: Fix an integer $k\geq 2$, and let $T$ be a real number with $T\geq 1.5$. Let $\cH=(V,\cE_2\cup \cE_3\cup\dots\cup\cE_k)$ be a non-uniform hypergraph with the vertex set $V$ and the set $\cE_i$ of edges of size $i=2,\ldots , k$. Suppose that $\cH$ has no $2$-cycles (regardless of sizes of edges), and neither contains $3$-cycles nor $4$-cycles consisting of $2$-element edges. If the average degrees $t_i^{i-1} := i |\cE_i|/ |V|$ satisfy that $t_i^{i-1} \leq T^{i-1} (\ln T)^{\frac{k-i}{k-1}}$ for $i= 2, \dots , k$, then there exists a constant $C_k > 0$, depending only on $k$, such that $\alpha(\cH)\geq C_k \frac{|V|}{T} (\ln T)^{\frac{1}{k-1}}$, where $\alpha(\cH)$ denotes the independence number of $\cH$. This extends results of Ajtai, Koml\'os, Pintz, Spencer and Szemer\'edi and Duke, R\"odl and the second author for uniform hypergraphs. As an application, we consider an anti-Ramsey type problem on non-uniform hypergraphs. Let $\cH=\cH(n;2,\ldots,\ell)$ be the hypergraph on the $n$-vertex set $V$ in which, for $s=2,\ldots,\ell$, each $s$-subset of $V$ is a hyperedge of $\cH$. Let $\Delta$ be an edge-coloring of $\cH$ satisfying the following: (a) two hyperedges sharing a vertex have different colors; (b) two hyperedges with distinct size have different colors; (c) a color used for a hyperedge of size $s$ appears at most $u_s$ times. For such a coloring $\Delta$, let $f_{\Delta}(n;u_2,\ldots,u_{\ell})$ be the maximum size of a subset $U$ of $V$ such that each hyperedge of $\cH[U]$ has a distinct color, and let $f(n;u_2,\ldots,u_{\ell}):=\min_{\Delta} f_{\Delta}(n;u_2,\ldots,u_{\ell}).$ We determine $f(n;u_2,\ldots,u_{\ell})$ up to a multiplicative logarithm factor.