Papa Sissokho

Embedding graphs with bounded degree in sparse pseudorandom graphs

AbstractIn this paper, we show the equivalence of somequasi-random properties for sparse graphs, that is, graphsG with edge densityp=|E(G)|/(2n)=o(1), whereo(1)→0 asn=|V(G)|→∞. Our main result (Theorem 16) is the following embedding result. For a graphJ, writeNJ(x) for the neighborhood of the vertexx inJ, and letδ(J) andΔ(J) be the minimum and the maximum degree inJ. LetH be atriangle-free graph and setdH=max{δ(J):J⊆H}. Moreover, putDH=min{2dH,Δ(H)}. LetC>1 be a fixed constant and supposep=p(n)≫n−1DH. We show that ifG is such that(i)degG(x)≤Cpn for allx∈V(G),(ii)for all 2≤r≤DH and for all distinct verticesx1, ...,xr ∈V(G),

The maximum size of a partial spread II

\left| {N_G (x_1 ) \cap \cdots \cap N_G (x_r )} \right| \leqslant Cnp^r

A canonical semi-classical star product

,(iii)for all but at mosto(n2) pairs {x1,x2} ⊆V(G),

The structure of the minimum size supertail of a subspace partition

\left\| {N_G (x_1 ) \cap N_G (x_2 )\left| { - np^2 } \right| = o(np_2 )} \right.

Partitions of finite vector spaces over GF(2) into subspaces of dimensions 2 and s

, then the number of labeled copies ofH inG is