Abstract. Vu, Wood and Wood showed that any finite set S in a characteristic zero integral domain can be mapped to Fp, for infinitely many primes p, while preserving finitely many algebraic incidences of S. In this note we show that the converse essentially holds, namely any small subset of Fp can be mapped to some finite algebraic extension of Q, while preserving bounded algebraic relations. This answers a question of Vu, Wood and Wood. We give several applications, in particular we show that for small subsets of Fp, the Szemerédi-Trotter theorem holds with optimal exponent 4/3, and we improve the previously best-known sumproduct estimate in Fp. We also give an application to an old question of Rényi. The proof of the main result is an application of elimination theory and is similar in spirit with the proof of the quantitative Hilbert Nullstellensatz.

The Graceful Tree Conjecture of Rosa from 1967 asserts that the vertices of each tree T of order n can be injectively labeled by using the numbers {1, 2, … , n} in such a way that the absolute differences induced on the edges are pairwise distinct. We prove the following relaxation of the conjecture for each > 0 and for all n > n 0 ( ). Suppose that (i) the maximum degree of T is bounded by O (n∕ log n), and (ii) the vertex labels are chosen from the set {1, 2, … , ⌈(1 + )n⌉}. Then there is an injective labeling of V(T) such that the absolute differences on the edges are pairwise distinct. In particular, asymptotically almost all trees on n vertices admit such a labeling. The proof proceeds by showing that a certain very natural randomized algorithm produces a desired labeling with high probability.

Abstract. Let r ≥ s ≥ 0 be integers and G be an r-graph. The higher inclusion matrix M r s (G) is a {0, 1}-matrix with rows indexed by the edges of G and columns indexed by the subsets of V (G) of size s: the entry corresponding to an edge e and a subset S is 1 if S ⊆ e and 0 otherwise. Following a question of Frankl and Tokushige and a result of Keevash, we define the rank-extremal function rex(n, t, r, s) as the maximum number of edges of an r-graph G having rk M r s (G) ≤ n s − t. For t at most linear in n we determine this function as well as the extremal r-graphs. The special case t = 1 answers a question of Keevash.

The projective norm graphs P(q, 4) introduced by Alon, Rónyai and Szabó are explicit examples of extremal graphs not containing K 4,7 . Ball and Pepe showed that P(q, 4) does not contain a copy of K 5,5 either for q ≥ 7, asymptotically improving the best lower bound for ex(n, K 5,5 ).We show that these results can not be improved, in the sense that P(q, 4) contains a copy of K 4,6 for infinitely many primes q.

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