Non-linear index coding outperforming the linear optimum
  The following source coding problem was introduced by Birk and Kol: a sender holds a word $x\in\{0,1\}^n$, and wishes to broadcast a codeword to $n$ receivers, $R_1,...,R_n$. The receiver $R_i$ is interested in $x_i$, and has prior \emph{side information} comprising some subset of the $n$ bits. This corresponds to a directed graph $G$ on $n$ vertices, where $i j$ is an edge iff $R_i$ knows the bit $x_j$. An \emph{index code} for $G$ is an encoding scheme which enables each $R_i$ to always reconstruct $x_i$, given his side information. The minimal word length of an index code was studied by Bar-Yossef, Birk, Jayram and Kol (FOCS 2006). They introduced a graph parameter, $\minrk_2(G)$, which completely characterizes the length of an optimal \emph{linear} index code for $G$. The authors of BBJK showed that in various cases linear codes attain the optimal word length, and conjectured that linear index coding is in fact \emph{always} optimal.   In this work, we disprove the main conjecture of BBJK in the following strong sense: for any $\epsilon > 0$ and sufficiently large $n$, there is an $n$-vertex graph $G$ so that every linear index code for $G$ requires codewords of length at least $n^{1-\epsilon}$, and yet a non-linear index code for $G$ has a word length of $n^\epsilon$. This is achieved by an explicit construction, which extends Alon's variant of the celebrated Ramsey construction of Frankl and Wilson.   In addition, we study optimal index codes in various, less restricted, natural models, and prove several related properties of the graph parameter $\minrk(G)$. 
Eyal Lubetzky, Uri Stav