The minrank of random graphs over arbitrary fields

Abstract

The minrank of a graph G on the set of vertices [n] over a field F is the minimum possible rank of a matrix M ∈ F n×n with nonzero diagonal entries such that M i,j = 0 whenever i and j are distinct nonadjacent vertices of G. This notion, over the real field, arises in the study of the Lovász theta function of a graph. We obtain tight bounds for the typical minrank of the binomial random graph G(n, p) over any finite or infinite field, showing that for every field F = F(n) and every p = p(n) satisfying n −1 ≤ p ≤ 1 − n −0.99 , the minrank of G = G(n, p) over F is Θ( n log(1/p) log n ) with high probability. The result for the real field settles a problem raised by Knuth in 1994. The proof combines a recent argument of Golovnev, Regev, and Weinstein, who proved the above result for finite fields of size at most n O(1) , with tools from linear algebra, including an estimate of Rónyai, Babai, and Ganapathy for the number of zero-patterns of a sequence of polynomials.

Perfect and nearly perfect separation dimension of complete and random graphs

“…with the following properties: (i) Each P i contains n parts of order n each, denoted by P i j , for j n = 1, …, . (ii) For any pair of distinct elements of n [ ] 2 , there is exactly one partition P i that contains both of them in the same part of P i . It is well-known that if n is a prime power, then there is a finite affine plane of order n [14].…”

“…In geometric terminology, the P i j , are called lines and the P i are called parallel classes. In our proof we will view each P i j , not just as a set, but as an ordered set of n elements of n [ ] 2 . In Figure 1 we list an affine plane of order 4 viewed in this way.…”

“…By induction we may assume the existence of a multiset  consisting of elements of S r , of size    g r = ( ) that is both a perfect separating family of K r and a perfect 3-sequence covering array of r [ ]. We must show that there exists a multiset  * of elements of S r 2 such that  ⋅   g r r g r * = ( ) = 2 (1 + ) ( ) 2 and such that  * is both a perfect separating family of K r 2 and a perfect 3-sequence covering array of r [ ] 2 .…”

“…The lemma's proof is very similar to the proof of the construction given in [21], but we repeat it for completeness (note, however, that the proof that  * is a perfect separating family given later is more involved and is not needed in the proof given in [21]). So, let a b c , , be three distinct integers in r [ ] 2 . We must prove that a b c , , is a subsequence of precisely  ∕ ∕   g r * 6 = ( ) 6 2 elements of  *.…”

Lower Bounds on the Time/Memory Tradeoff of Function Inversion

“…] is more involved since T might depend on Y i . 3 Yet since f is a random function, a simple counting argument yields that for any (fixed and independent of f ) function g:…”