On some topological and combinatorial lower bounds on chromatic number of Kneser type hyper graphs

In this paper, we prove a generalization of a conjecture of Erd\"{o}s, about the chromatic number of certain Kneser-type hypergraphs. For integers $n,k,r,s$ with $n\ge rk$ and $2\le s\le r$, the $r$-uniform general Kneser hypergraph $\mbox{KG}^r_s(n,k)$, has all $k$-subsets of $\{1,\dots,n\}$ as the vertex set and all multi-sets $\{A_1,\dots, A_r\}$ of $k$-subsets with $s$-wise empty intersections as the edge set. The case $r=s=2$, was considers by Kneser \cite{K} in 1955, where he conjectured that its chromatic number is $n-2(k-1)$. This was finally proved by Lov\'asz \cite{L} in 1978. The case $r>2$ and $s=2$, was considered by Erd\"os in 1973, and he conjectured that its chromatic number is $\left\lceil\frac{n-r(k-1)}{r-1}\right\rceil$. This conjecture was proved by Alon, Frankl and Lov\'asz \cite{AFL} in 1986. The case where $s>2$, was considered by Sarkaria \cite{S} in 1990, where he claimed to prove a lower bound for its chromatic number which generalized all previous results. Unfortunately, an error was found by Lange and Ziegler \cite{Z'} in 2006 in the induction method of Sarkaria on the number of prime factors of $r$, and Sarkaria's proof only worked when $s$ is less than the smallest prime factor of $r$ or $s=2$. In this paper, by applying the $\mathbb Z_p$-Tucker lemma of Ziegler \cite{Z} and Meunier \cite{M}, we finally prove the general Erd\"{o}s conjecture and prove the claimed result of Sarkaria for any $2\le s\le r$. We also provide another proof of a special case of this result, using methods similar to those of Alon, Frankl, and Lov\'asz \cite{AFL} and compute the connectivity of certain simplicial complexes that might be of interest in their own right.