TY - JOUR

T1 - Bipartite algebraic graphs without quadrilaterals

AU - Bukh, Boris

AU - Jiang, Zilin

N1 - Funding Information:
The authors were supported in part by the U.S. taxpayers through NSF grant DMS-1301548. In addition, the first author was also partly supported by a Sloan Fellowship. The second author would like to thank Hong Wang for her suggestion on the choice of terminology in algebraic geometry. We thank the referees for suggestions that helped to improve the exposition.
Funding Information:
The authors were supported in part by the U.S. taxpayers through NSF grant DMS-1301548 . In addition, the first author was also partly supported by a Sloan Fellowship. The second author would like to thank Hong Wang for her suggestion on the choice of terminology in algebraic geometry. We thank the referees for suggestions that helped to improve the exposition.
Publisher Copyright:
© 2018 Elsevier B.V.

PY - 2018/6

Y1 - 2018/6

N2 - Let Ps be the s-dimensional complex projective space, and let X,Y be two non-empty open subsets of Ps in the Zariski topology. A hypersurface H in Ps×Ps induces a bipartite graph G as follows: the partite sets of G are X and Y, and the edge set is defined by u¯∼v¯ if and only if (u¯,v¯)∈H. Motivated by the Turán problem for bipartite graphs, we say that H∩(X×Y) is (s,t)-grid-free provided that G contains no complete bipartite subgraph that has s vertices in X and t vertices in Y. We conjecture that every (s,t)-grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in y¯ is bounded by a constant d=d(s,t), and we discuss possible notions of the equivalence. We establish the result that if H∩(X×P2) is (2,2)-grid-free, then there exists F∈C[x¯,y¯] of degree ≤2 in y¯ such that H∩(X×P2)={F=0}∩(X×P2). Finally, we transfer the result to algebraically closed fields of large characteristic.

AB - Let Ps be the s-dimensional complex projective space, and let X,Y be two non-empty open subsets of Ps in the Zariski topology. A hypersurface H in Ps×Ps induces a bipartite graph G as follows: the partite sets of G are X and Y, and the edge set is defined by u¯∼v¯ if and only if (u¯,v¯)∈H. Motivated by the Turán problem for bipartite graphs, we say that H∩(X×Y) is (s,t)-grid-free provided that G contains no complete bipartite subgraph that has s vertices in X and t vertices in Y. We conjecture that every (s,t)-grid-free hypersurface is equivalent, in a suitable sense, to a hypersurface whose degree in y¯ is bounded by a constant d=d(s,t), and we discuss possible notions of the equivalence. We establish the result that if H∩(X×P2) is (2,2)-grid-free, then there exists F∈C[x¯,y¯] of degree ≤2 in y¯ such that H∩(X×P2)={F=0}∩(X×P2). Finally, we transfer the result to algebraically closed fields of large characteristic.

KW - Algebraic graph

KW - Quadrilateral-free graph

KW - Turán number

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U2 - 10.1016/j.disc.2018.03.005

DO - 10.1016/j.disc.2018.03.005

M3 - Article

AN - SCOPUS:85044083721

SN - 0012-365X

VL - 341

SP - 1597

EP - 1604

JO - Discrete Mathematics

JF - Discrete Mathematics

IS - 6

ER -