The p-ary code associated with the incidence structure of points and t-spaces in a projective space PG (m,q) , where q=p^h , is the 𝔽_p -subspace generated by the incidence vectors of the blocks of this design. The dual of this code consists of all vectors orthogonal to every codeword of the original code. The minimum weight of this dual code has been studied since the 1970s; the currently best known lower bound is from 2002 by Bagchi and Inamdar. In this paper we provide a substantial improvement of this bound in the case when h>1 and m, p >2 . The main ingredient of the proof is a lower bound on the size of multisets of AG(m,q) meeting each line in 0 p points. We prove this, by generalising the polynomial argument in [S. Ball, A. Blokhuis, A. Gács, P. Sziklai, Zs. Weiner: On linear codes whose weights and length have a common divisor, Adv. Math., 211 (2007), 94–104] from (ordinary) planar point sets to multisets in higher dimensional spaces. While the sharpness of the planar result for point sets is in general not known, we can show that our result for multisets is sharp for every h,m,p≥ 2 .
A bipartite graph B is called a brace if it is connected and every matching of size at most two in B is contained in some perfect matching of B. A conformal cross over some cycle C is a pair of disjoint paths P_1 , P_2 which are internally disjoint from C, the endpoints of each path separate the endpoints of the other path on C, and both C∪ P_1∪ P_2 and B-(V(C)∪ V(P_1)∪ V(P_2)) have a perfect matching. We show that if C is a 4-cycle in a brace B, then C has a a conformal cross if and only if B contains K_3,3 as a matching minor. This result implies a polynomial time algorithm which solves the 2-linkage problem for alternating paths in bipartite graphs with perfect matchings.
The modern theory of homogeneous structures begins with the work of Roland Fraïssé. The theory developed in the last seventy years is placed in the border area between combinatorics, model theory, algebra, and analysis. We turn our attention to its combinatorial pillar, namely, the work on the classification of structures for given homogeneity types, and focus onto the homomorphism homogeneous ones, introduced in 2006 by Cameron and Nešetřil. An oriented graph is called homomorphism homogeneous if every homomorphism between finite induced subgraphs extends to an endomorphism. In this paper we present a complete classification of the countable homomorphism homogeneous oriented graphs. Among these we identify those that are polymorphism homogeneous. Here an oriented graph is called polymorphism homogeneous if each of its finite powers is homomorphism homogeneous.
This paper considers some different measures for how additively structured a convex set can be. The main result gives a construction of a convex set A containing Ω(|A|^3/2) three-term arithmetic progressions.
We show that if the ground set of a matroid can be partitioned into k≥ 2 bases, then for any given subset S of the ground set, there is a partition into k bases such that the sizes of the intersections of the bases with S may differ by at most one. This settles the matroid equitability conjecture by Fekete and Szabó (Electron. J. Comb. 2011) in the affirmative. We also investigate equitable splittings of two disjoint sets S_1 and S_2 , and show that there is a partition into k bases such that the sizes of the intersections with S_1 may differ by at most one and the sizes of the intersections with S_2 may differ by at most two; this is the best one can hope for arbitrary matroids. We also derive applications of this result to matroid-constrained fair division problems. We show that there exists a matroid-constrained allocation that is envy-free up to one item if the valuations are identical and tri-valued additive. We also show that for bi-valued additive valuations, there exists a matroid-constrained allocation that provides everyone their maximin share.
