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- W2912741917 abstract "A family $mathcal Fsubset {[n]choose k}$ is $U(s,q)$ of for any $F_1,ldots, F_sin mathcal F$ we have $|F_1cupldotscup F_s|le q$. This notion generalizes the property of a family to be $t$-intersecting and to have matching number smaller than $s$. In this paper, we find the maximum $|mathcal F|$ for $mathcal F$ that are $U(s,q)$, provided $n>C(s,q)k$ with moderate $C(s,q)$. In particular, we generalize the result of the first author on the Erdős Matching Conjecture and prove a generalization of the Erdős-Ko-Rado theorem, which states that for $n> s^2k$ the largest family $mathcal Fsubset {[n]choose k}$ with property $U(s,s(k-1)+1)$ is the star and is in particular intersecting. (Conversely, it is easy to see that any intersecting family in ${[n]choose k}$ is $U(s,s(k-1)+1)$.) We investigate the case $k=3$ more thoroughly, showing that, unlike in the case of the Erdős Matching Conjecture, in general there may be $3$ extremal families." @default.
- W2912741917 created "2019-02-21" @default.
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- W2912741917 date "2019-01-26" @default.
- W2912741917 modified "2023-09-27" @default.
- W2912741917 title "Beyond the ErdH{o}s Matching Conjecture" @default.
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