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Beyond the Erdős-Ko-Rado theorem. (English) Zbl 0742.05080
$$[n]$$ denotes the set $$\{1,2,\ldots,n\}$$; for any set $$S$$, $$\binom{S}{k}$$ denotes the set of its $$k$$-subsets; $$n_r(k,t)$$ is defined to be $$(k - t+1)\left(2+ \frac{t-1}{r+1}\right)$$; $$\mathcal A_r$$ is the set $$\left\{F\in \binom{[n]}{k}: | F\cap[t+2r]|\geq t+r\right\}$$.
A $$t$$-intersecting family over $$[n]$$ is a set of subsets of $$[n]$$ that intersect pairwise in at least $$t$$ points.
The first author has conjectured that: Conjecture 2.1. If $$\mathcal F$$ is of maximum cardinality, then $${\mathcal F}={\mathcal A}_r$$ for some $$r$$.
“We prove Conjecture 2.1 for $$n>(k-t+1)c\sqrt{t/\log t}:$$
Theorem 2.4. Suppose that $$\mathcal F$$ is a $$t$$-intersecting family over $$[n]$$ of maximal cardinality. Suppose further that $$n$$ is in the range $$n_r(k,t)\leq n\leq(k-t+1)\left(2+\frac{t-1}{r}\right)$$, and $$t\geq 1+cr(r+1)/(1+\log r)$$. Then $$\mathcal F$$ is ismorphic to $${\mathcal A}_r$$ (or to $${\mathcal A}_{r+1}$$ in the case $$n=n_r(k,t))$$.
The proof is elementary. It uses the so-called ‘shifting’ operation, introduced by P. Erdős, Chao Ko and R. Rado in “Intersection theorems for systems of finite sets” [Q. J. Math., Oxf. II. Ser. 12, 313–320 (1961; Zbl 0100.01902)]. We follow the line of the first author, “The Erdős-Ko-Rado Theorem is true for $$n = ckt$$” [Combinatorics, Keszthely 1976, Colloq. Math. Soc. János Bolyai 18, 365–375 (1978; Zbl 0401.05001)], where several properties of the shifted families were proved”.

##### MSC:
 05D05 Extremal set theory 05A05 Permutations, words, matrices
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##### References:
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