Clique number of xor-powers of Kneser graphs
Zoltán Füredi, András Imolay, Ádám Schweitzer
TL;DR
The paper analyzes the clique number $f_\ell(n,k)$ of the xor-product of $\ell$ copies of Kneser graphs $KG(n,k)$, establishing tight bounds for the $k=1$ case and broad growth-rate results for general $k$ as $n$ grows. It introduces the semi-intersecting family framework to translate clique problems into extremal set theory questions, and develops both combinatorial (Bollobás-type) and algebraic (over $\mathbf{F}_2$) tools to bound $f_\ell(n,k)$. Key contributions include exact bounds for $f_\ell(n,1)$ when $\ell\ge3$, upper and lower bounds for $f_2(n,k)$ with the sharp magnitude of $c(k)$, and general upper/lower bounds for higher powers showing $f_\ell(n,k)$ may grow polynomially in $n$ with exponents depending on $\ell$. The work also provides explicit constructions via $\ell$-cores and fusion, proves an inductive upper bound across powers, and conjectures the lower-bound exponent is optimal, with several exact or near-tight cases discussed. The results push understanding of xor-products of combinatorial graphs and connect extremal set theory with graph products and coding-theoretic perspectives.
Abstract
Let $f_\ell(n, k)$ denote the clique number of the xor-product of $\ell$ isomorphic Kneser graphs KG(n,k). Alon and Lubetzky investigated the case of complete graphs as a coding theory problem and showed $f_\ell(n,1)\leq \ell n +1$. Imolay, Kocsis, and Schweitzer proved that $f_2(n,k)\leq n/k +c(k)$. Here, the order of magnitude of $c(k)$ is determined to be $Θ\left( k \binom{2k}{k} \right)$. By explicit constructions and by an algebraic proof, it is shown that $\ell n- 2\ell-1 \leq f_\ell(n,1)\leq \ell n-\ell+1$ (for all $n \geq 1$ and $\ell\geq 3$). Finally, it is proved that the order of magnitude of $f$ lies between $Ω\left(n^{\left\lfloor \log_2(\ell+1)\right\rfloor}\right)$ and $O\left(n^{\left\lfloor \frac{\ell+1}{2} \right\rfloor} \right)$ (as $\ell$, $k$ are given and $n\to \infty$). We conjecture that the lower bound gives the correct exponent.