One of the results in extremal combinatorics which has been proved and extended in many ways is the Erdös-Ko-Rado theorem [3]. A \(k\)-set system is a collection of subsets of \(\{1, 2,...,n\}\) of size \(k\). A \(k\)-set system is intersecting if its elements are pairwise non-disjoint.
Theorem 1. (Erdös, Ko, and Rado [3])
Let \(k\) and \(n\) be positive integers such that \(n \geq 2k\).
\(\ \ (1)\) If \(\mathfrak{F}\) is an intersecting \(k\)-set system on an \(n\)-set, then
\(|\mathfrak{F}| \leq \left(
\begin{array}{c}
n-1 \\
k-1 \\
\end{array}
\right)\).
\(\ \ (2)\) If \(n > 2k\), then \(\mathfrak{F}\) meets this bound if and only if \(\mathfrak{F}\) is the collection of all \(k\)-subsets containing a fixed element \(i \in \{1,...,n\}\).
Erdös-Ko-Rado theorem has been extended in many ways. In [6], Hsieh investigated the analogous problem for finite vector spaces. Let \(GF(q)\) denote a finite field with \(q\) elements.
Theorem 2. (Hsieh [6]) Let \(\mathfrak{F}\) be a family of \(k\)-dimensional subspaces of an \(n\)-dimensional vector space \(V\) over \(GF(q)\) such that the members of \(\mathfrak{F}\) intersect pairwise non-trivially.
\(\ \ (1)\) If \(n \geq 2k\), then
\(|\mathfrak{F}| \leq \left[
\begin{array}{c}
n-1 \\
k-1 \\
\end{array}
\right]_q\),
where the Gaussian coefficient
\(\left[
\begin{array}{c}
n-1 \\
k-1 \\
\end{array}
\right]_q
\)
denotes the number of \(k\)-dimensional subspaces of \(V\) containing a specific \(1\)-dimensional subspace of \(V\).
\(\ \ (2)\) If \(n> 2k\), then \(\mathfrak{F}\) meets this bound if and only if \(\mathfrak{F}\) is a family of \(k\)-dimensional subspaces of \(V\) containing a specific \(1\)-dimensional subspace of \(V\), see [6, Theorem 4.4].
Fix \(t \in \mathbb{N}\). In 1986, Frankl and Wilson [4] generalized Hsieh's results for the family \(\mathfrak{F}\) of \(k\)-dimensional subspaces of \(V\) such that, for any \(A,B \in \mathfrak{F}\), \(\dim(A\cap B)\geq t\).
Let \(\Omega\) be a finite set and \(G\) a permutation group on it. A subset \(A\) of \(G\) is intersecting if, for any \(\delta,~\tau \in A\), there exists \(x \in \Omega\) such that \(\delta(x)=\tau(x)\). As an intersecting set of the permutation group \(G\), we can name the stabilizer of a point. The Erdös-Ko-Rado theorem for permutation groups is finding the size of the largest intersecting set of \(G\). This problem goes back to 1977, see [2].
Theorem 3.Let \(\mathfrak{F}\) be an intersecting set of the symmetric group \(\mathbb{S}_n\).
\(\ \ (1)\) (Deza and Frank [2]) \(|\mathfrak{F}| \leq (n-1)!\).
\(\ \ (2)\) ([1,5,7,9]) \(\mathfrak{F} \) meets this bound if and only if \(\mathfrak{F}\) is a coset of the stabilizer of a point.
In [8], it has been proved that the size of intersection set of the permutation group \(PGL_2(q)\) acting on the projective line \(\mathbb{P}_q\), is at most \(q(q-1)\) and the only sets \(S\) that meet this bound are the cosets of the stabilizer of a point of \(\mathbb{P}_q\). Also, Guo and Wang [An Erdös-Ko-Rado theorem in general linear groups, arXiv:1107.3178] find the upper bound for the size of the intersecting set of \(GL_n(q)\) acting on \((GF(q))^n-\{0\}\). In the submitted paper, we study the Erdös-Ko-Rado theorem for \(SL_2(q)\) and \(GL_2(q)\) acting on \((GF(q))^2-\{0\}\) and \(PSL_2(q)\) acing on the projective line \(\mathbb{P}_q\). In this talk we concern the Erdös-Ko-Rado theorem for some
linear groups and symplectic groups.
References
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M. Deza, P. Frankl, On the maximal number of permutations with given maximal or minimal distance, J. Combin. The. Ser. A, 22 (1977), 352–360.
P. Erdös, C. Ko, R. Rado, Intersecting theorems for systems of finite sets, Quart. J. Math. Oxford Ser., 12, N2 (1961), 313–320.
P. Frankl, R.M. Wilson, The Erdös-Ko-Rado theorem for vector spaces, J. Combin. The. Ser. A, 43 (1986), 228–236.
C. Godsil, K. Meagher, A new proof of the Erdös-Ko-Rado theorem for intersecting families of permutations, European J. Combin., 30 (2009), 404–414.
W. N. Hsieh, Intersection theorems for systems of finite vector spaces, Discrete Math., 12 (1975), 1–16.
B. Larose, C. Malvenuto, Stable sets of maximal size in Kneser-type graphs, European J. Combin., 25, N5 (2004), 657–673.
K. Meagher, P. Spiga, An Erdös-Ko-Rado theorem for the derangement graph of \(PGL_2(q)\) acting on the projective line, J. Combin. The. Ser. A, 118 (2011), 532–544.
J. Wang, S.J. Zhang, An Erdös-Ko-Rado theorem in Coxeter groups, European J. Combin., 29 (2008), 1112–1115.
