# Zero-Sum $$K_m$$ Over $${{{\mathbb {Z}}}}$$ and the Story of $$K_4$$

• Yair Caro
• Amanda Montejano
Original Paper

## Abstract

We prove the following results solving a problem raised by Caro and Yuster (Graphs Comb 32:49–63, 2016). For a positive integer $$m\ge 2$$, $$m\ne 4$$, there are infinitely many values of n such that the following holds: There is a weighting function $$f:E(K_n)\rightarrow \{-1,1\}$$ (and hence a weighting function $$f: E(K_n)\rightarrow \{-1,0,1\}$$), such that $$\sum _{e\in E(K_n)}f(e)=0$$ but, for every copy H of $$K_m$$ in $$K_n$$, $$\sum _{e\in E(H)}f(e)\ne 0$$. On the other hand, for every integer $$n\ge 5$$ and every weighting function $$f:E(K_n)\rightarrow \{-1,1\}$$ such that $$|\sum _{e\in E(K_n)}f(e)|\le \left( {\begin{array}{c}n\\ 2\end{array}}\right) - 2h(n)$$, where $$h(n)=(n+1)$$ if $$n \equiv 0$$ (mod 4) and $$h(n)=n$$ if $$n \not \equiv 0$$ (mod 4), there is always a copy H of $$K_4$$ in $$K_n$$ for which $$\sum _{e\in E(H)}f(e)=0$$, and the value of h(n) is sharp.

## Keywords

Zero-sum Ramsey theory Complete subgraphs Pell equation

## Mathematics Subject Classification

05C55 05C15 11D09

## Notes

### Acknowledgements

We would like to thank our colleague Florian Luca for some fruitful discussions concerning some results of this work. We also thank the anonymous referees for their suggestions and comments that helped improving the final presentation of this paper. Adriana Hansberg was partially supported by PAPIIT IA103217, PAPIIT IN111819 and CONACyT project 219775. Amanda Montejano was partially supported by PAPIIT IN114016, PAPIIT IN116519 and CONACyT project 219827. Finally, we would like to acknowledge the support from Center of Innovation in Mathematics, CINNMA A.C.

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© Springer Japan KK, part of Springer Nature 2019

• Yair Caro
• 1