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We say that the families $mathcal F_1,ldots, mathcal F_{s+1}$ of $k$-element subsets of $[n]$ are cross-dependent if there are no pairwise disjoint sets $F_1,ldots, F_{s+1}$, where $F_iin mathcal F_i$ for each $i$. The rainbow version of the ErdH os Matching Conjecture due to Aharoni and Howard and independently to Huang, Loh and Sudakov states that $min_{i} |mathcal F_i|le maxbig{{nchoose k}-{n-schoose k}, {(s+1)k-1choose k}big}$. In this paper, we prove this conjecture for $n>3e(s+1)k$ and $s>10^7$. One of the main tools in the proof is a concentration inequality due to Frankl and the author.
For a collection $mathbf{G}={G_1,dots, G_s}$ of not necessarily distinct graphs on the same vertex set $V$, a graph $H$ with vertices in $V$ is a $mathbf{G}$-transversal if there exists a bijection $phi:E(H)rightarrow [s]$ such that $ein E(G_{phi(e)}
The ErdH{o}s-Faber-Lov{a}sz conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this paper, we prove this conjecture for every large $n$. We also provide stabili
An $r$-uniform hypergraph ($r$-graph for short) is called linear if every pair of vertices belong to at most one edge. A linear $r$-graph is complete if every pair of vertices are in exactly one edge. The famous Brown-ErdH{o}s-Sos conjecture states t
We prove that for any $varepsilon>0$, for any large enough $t$, there is a graph $G$ that admits no $K_t$-minor but admits a $(frac32-varepsilon)t$-colouring that is frozen with respect to Kempe changes, i.e. any two colour classes induce a connected
We prove the well-known Brown-ErdH{o}s-Sos Conjecture for hypergraphs of large uniformity in the following form: any dense linear $r$-graph $G$ has $k$ edges spanning at most $(r-2)k+3$ vertices, provided the uniformity $r$ of $G$ is large enough giv