For an n-dimensional vector space V over Fq, let Vk denote the family of all k-dimensional subspaces of V and let |Vk| = nkq. Let mq(n, k, s) be the maximum size of a family F Vk containing no s+1 members F1,…,Fs+1 whose sum is direct; equivalently, there do not exist F1,…,Fs+1∈F  such that .

There are two natural constructions yielding lower bounds for mq(n, k, s):

Aq(k, s)=V1k, where V1 is a fixed ((s+1)k-1)-dimensional subspace,

Bq(n, k, s )={F∈Vk :F∩V2≠{0}}, where V2 is a fixed s-dimensional subspace.

We conjecture that one of these two constructions is always extremal. More precisely, for all n ≥ (s+1)k,

mq(n, k, s)=max{(s+1)k - 1kq,  nkq - qks n - skq}.

This is a vector-space analogue of the famous Erdős Matching Conjecture, posed by Erdős in 1964, and remains one of the central open problems in extremal set theory.

When s = 1, the conjecture reduces to the vector-space Erdős--Ko--Rado theorem, proved by Frankl and Wilson in 1986. We verify the conjecture in the cases k=2, n = (s+1)k, and n ≥ (2s + 1)k + 1.

Generalizing the classical notion of cover-free families introduced by Erdős, Frankl, and Füredi, we say that a family F Vr is t-cover-free if there do not exist F, F1, … ,Ft∈F  such that .We establish a new connection between the vector-space Erdős Matching Conjecture and vector-space cover-free families, and determine, up to a lower-order term, the maximum size of a t-cover-free family in Vr  for all fixed q, t, r as n→∞. This extends a recent result of Shan and Zhou for the case t=2.

This is joint work with Baoyan Feng, Yulin Yang, and Chenyang Zhang.

上官冲，山东大学教授、博导，主要从事组合图论与信息科学交叉领域的研究。主持基金委青年、面上、海外优青项目，获国际组合数学及应用协会2020年度 Kirkman 奖。在数学综合期刊Peking Math. J、 Sci. China Math.，组合图论期刊J. Combin. Theory Ser. A、J. Combin. Theory Ser. B、SIAM J. Discrete Math.，理论计算机科学与编码理论会议和期刊STOC、FOCS、SIAM J. Comput.、IEEE Trans. Inform. Theory等发表论文30余篇。